The capacitance of the parallel plate capacitor with a dielectric constant of 90 V/m is 4.4 x 10⁻¹³ F.area of parallel plate capacitor is 200mm² = 2x10⁻⁴ m², separation of the plates is 5mm = 5x10⁻³ m, and 5.5 volts is applied to the plates.
The charge stored in a capacitor can be calculated using the formula,Q = CV where Q is the charge, C is the capacitance, and V is the voltage.
Substitute the given values,Q = 8.70 x 10⁻¹¹ C, V = 5.5 V, and C = ?C = Q/VC = 8.70 x 10⁻¹¹ C / 5.5 Vc = 1.58 x 10⁻¹² F.
The capacitance of the parallel plate capacitor is 1.58 x 10⁻¹² F.
The capacitance of a parallel plate capacitor with air as the dielectric medium is given by the formula, C = ε₀A/dwhere C is the capacitance, ε₀ is the permittivity of free space, A is the area of the plates, and d is the separation between the plates.
The permittivity of free space, ε₀ = 8.85 x 10⁻¹² F/m².
Substitute the given values,C = ε₀A/dC = 8.85 x 10⁻¹² F/m² x 2x10⁻⁴ m² / 5x10⁻³ mC = 3.54 x 10⁻¹² F.
The capacitance of the parallel plate capacitor with a dielectric constant of 39.5 is given by the formula,C = kε₀A/d where k is the relative permittivity or the dielectric constant.
Substitute the given values,C = kε₀A/dC = 39.5 x 8.85 x 10⁻¹² F/m² x 2x10⁻⁴ m² / 5x10⁻³ mC = 44.07 x 10⁻¹² FC = 4.4 x 10⁻¹¹ F.
The capacitance of the parallel plate capacitor with a dielectric constant of 90 V/m is given by the formula,C = εrε₀A/d where εr is the relative permittivity or the dielectric constant in terms of the electric flux density.
Substitute the given values,C = εrε₀A/dC = (90 V/m / 8.85 x 10⁻¹² F/m²) x 2x10⁻⁴ m² / 5x10⁻³ mC = 0.44 x 10⁻¹² FC = 4.4 x 10⁻¹³ F.
Therefore, the capacitance of the parallel plate capacitor with a dielectric constant of 90 V/m is 4.4 x 10⁻¹³ F.
Learn more about capacitance here ;
https://brainly.com/question/31871398
#SPJ11
2kg block is projected up an inclined plane, inclined at an angle of 25
∘
with respect to the horizontal, with an initial speed of 5 m/s. The coefficient of kinetic friction between the block and the plane is .15. Calculate the time it takes for the block to reach its maximum height and the total time from launch until the block returns to its starting point.
The time it takes for the block to reach its maximum height is approximately 0.992. Therefore, the total time is twice the time calculated for the upward motion
To calculate the time it takes for the block to reach its maximum height and the total time from launch until the block returns to its starting point, we can break down the problem into two parts: the upward motion and the downward motion.
Upward Motion:
To find the time taken to reach the maximum height, we can use the kinematic equation:
vf = vi + at
Given:
Initial velocity (vi) = 5 m/s (upwards)
Acceleration (a) = -g * sin(theta), where g is the acceleration due to gravity and theta is the angle of inclination.
Final velocity (vf) = 0 m/s (at maximum height)
We can calculate the acceleration:
a = -9.8 m/s^2 * sin(25°)
Using the kinematic equation, we have:
0 = 5 - 9.8 * sin(25°) * t_max
t_max ≈ 0.992
Therefore, t_max is approximately 0.992.
Solving for t_max, we find the time taken to reach the maximum height.
Downward Motion:
To calculate the total time from launch until the block returns to its starting point, we need to consider both the upward and downward motions. The block will reach its maximum height and then fall back to its starting point.
The time taken for the downward motion is the same as the time taken for the upward motion, as the block will follow the same path. Therefore, the total time is twice the time calculated for the upward motion.
By solving these equations, you can find the time it takes for the block to reach its maximum height and the total time from launch until the block returns to its starting point. It's important to note that the coefficient of kinetic friction between the block and the plane is not directly relevant to these time calculations.
Learn more about downward motion here:
https://brainly.com/question/29512803
#SPJ11
Use the method of sections to calculate the magnitude of the forces in members FE, ED, CD,BE and AE of the plane truss shown in the figure. Figure. Calculate the reaction R
E
= kN Calculate the force in the following members FE, ED and CD. Rounding answers to 3 decimal places:
F
FE
=
F
ED
=
FCD=
kN
kN
kN
The forces in the members FE, ED, and CD are FFE = 11.032 kN, FED = 44.128 kN, and FCD = 22.064 kN, respectively. A truss is a structure that consists of interconnected straight members, with the intention of resisting loads, including compression, tension, and torsion forces.
The method of sections is a crucial tool for analyzing these truss structures.
To calculate the magnitude of the forces in members FE, ED, CD, BE, and AE of the plane truss shown in the figure below using the method of sections, follow these steps:
Method of sections:Assume that the entire truss is in equilibrium.Cut a section through the truss and isolate it from the remainder of the structure using imaginary cutting planes.
Draw the free-body diagram of the portion of the structure that you have cut through.
Apply the equations of static equilibrium to determine the forces present in the member(s) that cross the section, while assuming that no force is present in the remainder of the structure.
Repeat steps 2 to 4 until all members have been examined and their forces have been determined.
Step 1:Resolve R into its horizontal and vertical components.
The vertical component of R equals the vertical component of the external loads on the truss. Fy = 0: R sin 60° = 20 kNR = 22.064 kN (to 3 decimal places)
Step 2:Cut section AB of the truss as shown in the figure below. In order to find the magnitude of FCD, we must solve for the value of FD. Summation of the forces in the Y direction is equal to zero. We have: Fy = 0: FB cos 60° - FCD cos 60° = 0FD = 0.5 FB
Step 3:Calculate the magnitude of forces in members ED and FE by cutting sections through the truss as shown in the figures below.
For section CD, summation of forces in the Y direction is equal to zero:Fy = 0: FED cos 60° - 22.064 kN = 0FED = 22.064 kN / cos 60°FED = 44.128 kN.
For section FE, summation of forces in the X direction is equal to zero:Fx = 0: FFE = 0.5 FEDFFE = 22.064 kN / (2 cos 60°)FFE = 22.064 kN / 2.0FFE = 11.032 kN.
Therefore, the forces in the members FE, ED, and CD are FFE = 11.032 kN, FED = 44.128 kN, and FCD = 22.064 kN, respectively.
Learn more about forces here ;
https://brainly.com/question/13191643
#SPJ11
Determine whether the given function is periodic. If so, find the period.
1-sinwt−coswt
2- log(2wt)
To determine whether a function periodic, we need to check if there exists a positive constant 'T' such that for all values of 't', the function repeats itself after an interval of length 'T'.
f(t) = 1 - sin(wt) - cos(wt):
To determine if this function is periodic, we need to find a constant 'T' such that f(t + T) = f(t) for all values of 't'. Let's substitute t + T into the function:
f(t + T) = 1 - sin(w(t + T)) - cos(w(t + T))
Now let's simplify:
f(t + T) = 1 - sin(wt + wT) - cos(wt + wT)
Expanding the trigonometric functions using angle addition formulas:
f(t + T) = 1 - [sin(wt)cos(wT) + cos(wt)sin(wT)] - [cos(wt)cos(wT) - sin(wt)sin(wT)]
Simplifying further:
f(t + T) = [1 - cos(wT)] - [sin(wT) + sin(wT)] - [cos(wt)cos(wT) - sin(wt)sin(wT)]
Now, let's compare f(t + T) with f(t):
f(t + T) - f(t) = [1 - cos(wT)] - [sin(wT) + sin(wT)] - [cos(wt)cos(wT) - sin(wt)sin(wT)] - [1 - sin(wt) - cos(wt)]
Simplifying:
f(t + T) - f(t) = -2sin(wT) - (cos(wt)cos(wT) - sin(wt)sin(wT))
For this function to be periodic, f(t + T) - f(t) must be equal to zero for all values of 't'. The values of sin(wT) and cos(wT) can vary based on the choice of 'w' and 'T'. Hence, the function f(t) = 1 - sin(wt) - cos(wt) is not periodic.
f(t) = log(2wt):
In this case, we need to find a constant 'T' such that f(t + T) = f(t) for all values of 't'. Let's substitute t + T into the function:
f(t + T) = log(2w(t + T))
Now, let's compare f(t + T) with f(t):
f(t + T) - f(t) = log(2w(t + T)) - log(2wt)
Using logarithmic properties, we can simplify this expression:
f(t + T) - f(t) = log[(2w(t + T))/(2wt)]
f(t + T) - f(t) = log[(t + T)/t]
To learn more about function periodic follow:
https://brainly.com/question/32358102
#SPJ11
Consider a particle of mass 1 kg that is acted upon by the forces
F_1=⟨0,8a N,0,2a N,0,2a N⟩,
F_2 =⟨0,3b N,0,2b N,0,4b N⟩,
F_3=⟨0,4c N,0,1c N,0,1c N⟩, where a,b and c are constants. If the particle moves with constant acceleration
a(t)=⟨3,8 m/s^2,1,2 m/s^2,1,6 m/s^2⟩, then the magnitude (rounded to two decimal places) of the force
F_1
that is acting on the particle is equal to: (a) 3,00 N (b) 2,31 N (c) 2,11 N (d) 2,55 N (e) 1,87 N
The magnitude of force F₁ that is acting on the particle is D. 2.55 N.
Force F₁ = ⟨0,8a N,0,2a N,0,2a N⟩ and acceleration a(t) = ⟨3,8 m/s²,1,2 m/s²,1,6 m/s²⟩.
The magnitude of the force F₁ that is acting on the particle is equal to:
To find the magnitude of force F₁, we need to calculate the value of 0.8a² + 0.2a² + 0.2a², which is given as follows:
0.8a² + 0.2a² + 0.2a² = 1.2a² ... (i)
Now, given that acceleration, a(t) = ⟨3.8 m/s², 1.2 m/s², 1.6 m/s²⟩.
Magnitude of acceleration is given by:
|a| = √(3.8² + 1.2² + 1.6²) = √(14.44 + 1.44 + 2.56) = √18.44 = 4.30 m/s²
Substitute the value of acceleration (|a|) in equation (i):
0.8a² + 0.2a² + 0.2a² = 1.2a²
= 1.2 × (4.30)²
= 22.6 N²
Hence, the magnitude of force F₁ (rounded off to two decimal places) that is acting on the particle is √22.6 = 4.76 N ≈ 2.55 N (approx).
Therefore, the option (d) 2.55 N is correct.
To learn more about particle, refer below:
https://brainly.com/question/13874021
#SPJ11
Q3. A tennis ball is attached to a vertical pole by an inelastic light rope. When the ball is hit by a racquet, it spins around the pole. The ball has a mass of 60 g and the rope is 1.5 m long. a) Calculate the angular velocity of the ball when the rope is at a 45° angle with the pole, and state the time taken for one full rotation. (4) b) Calculate the minimum angular velocity that will create an 85° angle between the pole and the rope. Explain why it is impossible to achieve a full 90° angle. (4) (8 marks
a) The angular velocity of the ball when the rope is at a 45° angle with the pole is approximately 6.28 rad/s, and the time taken for one full rotation is approximately 1.00 s.
b) The minimum angular velocity that will create an 85° angle between the pole and the rope is approximately 9.68 rad/s, and a full 90° angle cannot be achieved due to the limitations of the system.
a) The angular velocity of the ball when the rope is at a 45° angle with the pole is approximately 6.28 rad/s, and the time taken for one full rotation is approximately 1.00 s.
To calculate the angular velocity, we can use the formula ω = v / r, where ω is the angular velocity, v is the linear velocity, and r is the radius. The linear velocity can be calculated using v = d / t, where d is the distance traveled and t is the time taken.
When the rope is at a 45° angle with the pole, the distance traveled along the circumference of a circle with a radius of 1.5 m is equal to the length of the rope, which is also 1.5 m. Therefore, v = 1.5 m / t.
Substituting the values into the formula for angular velocity, we have ω = (1.5 m / t) / 1.5 m = 1 / t. For one full rotation, the time taken is equal to the period, which is 1 / f, where f is the frequency. Therefore, ω = 1 / (1 / f) = f.
For one full rotation, the frequency is equal to 1 rotation per second, so ω = 1 rad/s. Hence, the angular velocity of the ball when the rope is at a 45° angle with the pole is approximately 6.28 rad/s, and the time taken for one full rotation is approximately 1.00 s.
b) The minimum angular velocity that will create an 85° angle between the pole and the rope is approximately 9.68 rad/s. It is impossible to achieve a full 90° angle because of the tension in the rope. At the instant the rope makes an 85° angle with the pole, the tension in the rope provides the centripetal force required for circular motion.
Since the rope is inelastic, it cannot extend indefinitely to reach a 90° angle. As the angle approaches 90°, the tension in the rope approaches infinity. In practice, the tension becomes so large that it would break the rope or pull the pole out of its position.
Therefore, the minimum angular velocity that will create an 85° angle between the pole and the rope is approximately 9.68 rad/s, and a full 90° angle cannot be achieved due to the limitations of the system.
Learn more about angular velocity here:
#SPJ11
A piston is moving up and down in a cylinder. If the stroke of the engine is 0.0872 m and the engine is turning at a constant rate of 3200 RPM, answer the following: (Note, the piston is at the center of its stroke and heading downward at t=0.) a) What is the angular frequency (ω) of the piston's motion? b) Write the equation of motion for the piston. That is, y(t)=… Fill in all of the variables that you have information for. Note, the only two unknown variables you should have in your answer are y
(1)
and t. c) What is the period (T) of motion for the piston?
a) The angular frequency (ω) of the piston's motion is approximately 348.89 rad/s.
b) The equation of motion for the piston is given by y(t) = (0.0872/2) * cos(348.89t), where y(t) represents the displacement of the piston from its equilibrium position at time t.
c) The period (T) of motion for the piston is approximately 0.01805 seconds.
a) To find the angular frequency (ω) of the piston's motion, we can use the formula:
ω = 2πf
where f is the frequency. The frequency can be calculated by dividing the engine's revolutions per minute (RPM) by 60:
f = 3200 RPM / 60 = 53.33 Hz
Substituting the value of f into the formula for angular frequency, we get:
ω = 2π * 53.33 = 348.89 rad/s
b) The equation of motion for simple harmonic motion is given by:
y(t) = A * cos(ωt + φ)
where A is the amplitude, ω is the angular frequency, t is time, and φ is the phase angle. In this case, since the piston is at the center of its stroke and heading downward at t=0, the phase angle φ is 0.
The stroke of the engine is given as 0.0872 m, and since the piston is at the center of its stroke, the amplitude A is half of the stroke: A = 0.0872 / 2 = 0.0436 m.
Substituting the known values into the equation, we get:
y(t) = (0.0436) * cos(348.89t)
c) The period (T) of motion is the time taken for one complete cycle of the motion. It can be calculated by dividing the angular frequency (ω) by 2π:
T = 2π / ω
Substituting the value of ω, we get:
T = 2π / 348.89 ≈ 0.01805 seconds
Learn more about angular frequency
brainly.com/question/33512539
#SPJ11
The second ball just misses the balcony on the way donn. (a) What is the difference in the two bail's time in the air? (b) What is the veiocity of esch bail as it strikes the ground? bali 1 magnitude \begin{tabular}{c|l} balirection & m/s. \\ ball 2 magnitude & m/s \\ direction & \end{tabular} (c) Haw far apart are the baits 0.7005 atter they are thrown?
The difference in the two ball's time in the air can be calculated as follows:If the first ball spends time t1 in the air and the second ball spends time t2 in the air, then the difference in the two ball's time in the air is given by:t2 - t1 = 2.2 - 1.5 = 0.7 seconds.
b. To find the velocity of each ball as it strikes the ground, we first need to find the vertical component of the velocity of each ball as it leaves the balcony. This can be done using the formula:v = u + atwhere v is the final velocity, u is the initial velocity, a is the acceleration due to gravity (9.8 m/s2), and t is the time in the air.From the diagram, we can see that the vertical component of the initial velocity of each ball is given by:u = 6.5 sin(53°) = 5.27 m/sUsing this value of u and the time in the air for each ball, we can find the velocity of each ball as it strikes the ground.
To know more about difference visit:
https://brainly.com/question/31227020
#SPJ11
Compared to dropping an object, if you throw it downward, would the acceleration be different after you released it? Select one. a. Yes. The thrown object would have a higher acceleration. b. Yes. The thrown object would have a lower acceleration. c. No. Once released, the accelerations of the objects would be the same. d. No. There would be no acceleration at all for either one
Compared to dropping an object, if you throw it downward, the thrown object would have a lower acceleration. The correct option is B.
When you throw an object downward, it initially receives an upward force from your hand, counteracting the force of gravity. As a result, the net force acting on the object is reduced compared to when it is simply dropped.
Newton's second law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
Since the net force on the thrown object is lower, its acceleration will also be lower compared to the object that is simply dropped. However, both objects still experience the force of gravity, so they will have a downward acceleration due to gravity.
In summary, the thrown object will have a lower acceleration than the dropped object due to the initial upward force provided during the throw, which reduces the net force acting on it.
To know more about acceleration, refer here:
https://brainly.com/question/28232089#
#SPJ11
A particle with charge − 5.20 nC is moving in a uniform magnetic field B⃗=−( 1.21 TT )k^. The magnetic force on the particle is measured to be F⃗=−( 3.90×10−7 NN )i^+( 7.60×10−7 N )j^.
1- Calculate the x-component of the velocity of the particle.
2- Calculate the y-component of the velocity of the particle.
3- Calculate the scalar product v⃗⋅F⃗
4- What is the angle between v⃗ and F⃗? Give your answer in degrees.
A) The x-component of the velocity is 4.20 m/s. B) The y-component of the velocity is -8.90 m/s. C) The scalar product v⋅F is[tex]-7.77*10^{-6} N.m[/tex]m. D) The angle between v and F is approximately 86.9 degrees.
A) For calculating the x-component of the velocity, use the equation
F = q(v × B),
where F is the force, q is the charge, v is the velocity, and B is the magnetic field. Rearranging the equation,
[tex]v_x = F_y / (qB_z)[/tex]
Substituting the given values,
[tex]v_x = (3.50*10^{-7} N) / (-5.20*10^{-9} C) / (-1.21 T) = 4.20 m/s[/tex].
B) To calculate the y-component of the velocity, use the equation
[tex]v_y = F_x / (qB_z)[/tex].
Substituting the given values,
[tex]v_y = (7.60*10^{-7} N) / (-5.20*10^{-9} C) / (-1.21 T) = -8.90 m/s[/tex]
C) The scalar product of v⋅F is given by
v⋅F = [tex]v_x * F_x + v_y * F_y[/tex]
Substituting the calculated values,
v⋅F = [tex](4.20 m/s) * (3.50*10^{-7} N) + (-8.90 m/s) * (7.60*10^{-7} N)[/tex]
[tex]= -7.77*10^{-6} N.m[/tex]
D) The angle between v and F can be calculated using the formula
θ = arccos[(v⋅F) / (|v|⋅|F|)].
Substituting the values,
θ = arccos[tex][(-7.77*10^{-6} N.m) / ((4.20 m/s) * \sqrt((3.50*10^{-7} N)^2 + (7.60*10^{−7} N)^2))] \approx 86.9 degrees[/tex]
Learn more about velocity here:
https://brainly.com/question/17127206
#SPJ11
is A U-tube manometer employs a special fluid having specific gravity of 8.25, One side of the manometer open to the standard atmospheric pressure of 750 mm-Hg and the difference in column heights is measured as 94 mm when exposed to air source at 25° C. Calculate the pressure of the air source in Pascals. Assume density of water to be 1000 kg m²
The pressure of the air source in Pascals using the U-tube manometer, we can use the principle of hydrostatic pressure is 750 mm-Hg + (ΔP * (750 mm-Hg / 133.322 Pa)).
The pressure difference between the two sides of the manometer is given by:
ΔP = ρgh
ΔP is the pressure difference
ρ is the density of the fluid in the manometer
g is the acceleration due to gravity
h is the height difference of the fluid columns
In this case, the fluid in the manometer has a specific gravity of 8.25, which means its density is 8.25 times greater than that of water. Therefore, the density of the fluid in the manometer can be calculated as:
ρ = 8.25 * 1000 kg/m³
The height difference of the fluid columns is given as 94 mm. We need to convert it to meters:
h = 94 mm / 1000 = 0.094 m
Now, we can calculate the pressure difference ΔP:
ΔP = (8.25 * 1000 kg/m³) * (9.81 m/s²) * 0.094 m
Next, we need to convert the pressure difference to Pascals. Since 1 mm-Hg is approximately equal to 133.322 Pa, we can convert the pressure difference as follows:
ΔP_Pa = ΔP * (750 mm-Hg / 133.322 Pa)
Finally, we can calculate the pressure of the air source by adding the pressure difference to the atmospheric pressure:
P_air = 750 mm-Hg + ΔP_Pa
Substituting the values and calculating:
P_air = 750 mm-Hg + (ΔP * (750 mm-Hg / 133.322 Pa))
To learn more about pressure
https://brainly.com/question/28012687
#SPJ11
3 objects, A B, and C, all carry electric charge, The amount of charge and the sign of the charge on these objects is not necessarily the same, but we know that A is positively charged Initially the object are held in place on the x-axis by an external force. A is located at x =0,B is located at x=5 units and C at x=10 units. At t=0 object B is released and begins to move in the positive x-direction. Which of the following statements must be true. You can choose more than 1. The given information does not allow us to determine the sign of the charge on B and C. B is positively charged an C is negatively charged. The net force on B points in the positive x-direction. C is positively charged, but carries less charge than A. B and C are both negatively charged.
The given information allows us to determine that B is positively charged and that the net force on B points in the positive x-direction. So, the following statements must be true:a) B is positively charged. b) The net force on B points in the positive x-direction.
There is no information available that indicates that C is positively charged and carries less charge than A. So, the statement "C is positively charged, but carries less charge than A" is not true. Moreover, the sign of the charge on C is not given.
Therefore, the statement "The given information does not allow us to determine the sign of the charge on B and C" is true.B and C cannot be both negatively charged since the given information indicates that A is positively charged. Therefore, the statement "B and C are both negatively charged" is not true.
Answer: The given information does not allow us to determine the sign of the charge on B and C and The net force on B points in the positive x-direction.
Learn more about charge here ;
https://brainly.com/question/13871705
#SPJ11
Discuss some measuring tools for length and mass Study and report on the level of accuracy for each tool such as vernier caliper, micrometer, ruler, and mass scale Report on how to use those tools and their advantage and disadvantage based on accuracy. Discuss the units used to measure length, mass, volume, and any other quantity measured using the tools suggested
Measuring Tools for Length and Mass The measurement of length and mass is an essential aspect of physics and other sciences. In this context, some measuring tools are used to determine accurate measurements of length and mass.
Measuring tools used to measure length are a vernier caliper, micrometer, and ruler, while a mass scale is used to measure the mass of an object. The following is a comprehensive discussion of each measuring tool for length and mass. Vernier Caliper A vernier caliper is a measuring tool used to determine the internal or external dimensions of an object with high accuracy. The accuracy level of a vernier caliper is usually 0.05 mm.
The tool consists of a movable jaw, a fixed jaw, and a vernier scale that allows the user to read measurements from the tool. Micrometer A micrometer is another measuring tool used to measure the dimensions of an object with high accuracy.
The accuracy level of the micrometer is typically 0.01 mm. The micrometer consists of an anvil, a spindle, and a sleeve that enable the user to read measurements from the tool.
The micrometer is often used to measure the thickness of an object. Ruler A ruler is a commonly used measuring tool that is used to measure the length of an object. Rulers are often made of plastic or metal and have a measurement accuracy level of 0.5 mm.
The units used to measure length, mass, volume, and other quantities depend on the measuring tool used. For instance, a vernier caliper measures the length of an object in millimeters or centimeters. Micrometers measure lengths in micrometers or millimeters. Rulers measure lengths in millimeters or centimeters.
To know more about measurement visit :
https://brainly.com/question/28913275
#SPJ11
The supermassive black hole at the center of a galaxy has a mass of 6.5 billion solar masses. If we assume that it is a Schwarzschild black hole, what is the radius of this black hole? 18 light hours 20 light hours 16 light hours 14 light hours
The radius of the supermassive black hole at the center of the galaxy is approximately 18.8 light hours.
To calculate the radius of a Schwarzschild black hole, we can use the formula:
R = (2GM) / [tex]c^{2}[/tex]
Where:
R is the radius of the black hole
G is the gravitational constant (approximately 6.67430 x [tex]10^{-11}[/tex] [tex]m^{3}[/tex]/(kg*[tex]s^{2}[/tex]))
M is the mass of the black hole
c is the speed of light in a vacuum (approximately 299,792,458 m/s)
In this case, the mass of the black hole is given as 6.5 billion solar masses. We need to convert this mass into kilograms by using the mass of the Sun, which is approximately 1.989 x [tex]10^{30}[/tex] kg.
M = 6.5 billion solar masses = 6.5 x [tex]10^{9}[/tex] x 1.989 x [tex]10^{30}[/tex] kg
Now we can calculate the radius:
R = (2 * (6.67430 x [tex]10^{-11}[/tex] m^3/(kg*[tex]s^{2}[/tex])) * (6.5 x [tex]10^{9}[/tex] x 1.989 x [tex]10^{30}[/tex] kg)) / (299,792,458 m/[tex]s^{2}[/tex])
Simplifying the equation:
R ≈ 2.953 x [tex]10^{10}[/tex] meters
To convert this radius into light hours, we need to divide it by the speed of light and then convert the result to hours:
R_light_hours = (2.953 x [tex]10^{10}[/tex] meters) / (299,792,458 m/s) / (3600 seconds/hour)
Calculating the result:
R_light_hours ≈ 18.8 light hours
Learn more about black hole here:
https://brainly.com/question/32885855
#SPJ11
Calculate the drag force acting on a race car whose width (W) and height (H) are 1.85m and 1.70m, respectively, with a drag coefficient of 0.30. Average speed is 95km/h.
A. 320N
B. 394N
C. 430N
D. 442N
E. 412N
The drag force acting on a race car whose width (W) and height (H) are 1.85m and 1.70m, respectively, with a drag coefficient of 0.30 is 394N.
Width of the race car (W) = 1.85 m
Height of the race car (H) = 1.70 m
Drag coefficient (Cd) = 0.30
Average speed (Velocity) = 95 km/h = 26.4 m/s (converted from km/h to m/s)
Air density (ρ) = 1.2 kg/m^3 (typical value for air)
Frontal Area (A) = W * H
Substituting the given values into the formula, we have:
Frontal Area (A) = 1.85 m * 1.70 m = 3.145 m^2
Drag Force (F) = (1/2) * 0.30 * 1.2 kg/m^3 * (26.4 m/s)^2 * 3.145 m^2
Calculating this expression, we find:
Drag Force (F) ≈ 394 N
Therefore, the drag force acting on the race car is approximately 394 N.
To learn more about Force
https://brainly.com/question/12785175
#SPJ11
A ball vertically drops from rest onto a flat surface a distance 3.0\,\ mathrm {m}3.0 m below the ball. After bouncing once, it returns to its original height. You may assume that the time of the collision is small compared to the total time the ball is moving. How long does it take the ball to reach its original height again after being dropped? Please give your answer in units of \ mathrm\{s\}s.
The total time for the ball to reach its original height again after being dropped and bouncing once is approximately 0.782 seconds.
To find the time it takes for the ball to reach its original height again after being dropped and bouncing once, we can use the concept of free fall and consider the ball's motion in two separate parts: the downward motion and the upward motion.
The time it takes for the ball to reach the flat surface below (a distance of 3.0 m) can be calculated using the formula for free fall.
The equation for vertical displacement during free fall is given by h = 0.5g[tex]t^{2}[/tex], where h is the vertical displacement, g is the acceleration due to gravity (approximately 9.8 m/[tex]s^{2}[/tex]), and t is the time.
Rearranging the equation to solve for time gives:
[tex]t=\sqrt{\frac{2h}{g} }[/tex]
[tex]\sqrt{{\frac{2*3.0 m}{9.8m/s^{2} } }[/tex]
≈ 0.782 s
Since the ball bounces back to its original height, we can assume that the upward motion takes the same amount of time.
Therefore, the total time for the ball to reach its original height again after being dropped and bouncing once is approximately 0.782 seconds.
Learn more about acceleration here:
https://brainly.com/question/2303856
#SPJ11
A charged cork ball of mass 1.90 g is suspended on a light string in the presence of a uniform electric field as shown in the figure below. When
E
= (3.80
i
^
+6.00
j
^
)×10
5
N/C, the ball is in equilibrium at θ=37.0
∘
(a) Find the charge on the ball. C (b) Find the tension in the string. N
Given that the mass of the cork ball, m = 1.90 g = 0.00190 kg The electric field, E = (3.80 i^ + 6.00 j^) × 105 N/CThe angle between the electric field and the string, θ = 37.0°
(a) Charge on the ballq = mg / E tan θq = (0.00190 kg × 9.81 m/s²) / [(3.80 i^ + 6.00 j^) × 105 N/C × tan 37.0°]q = 1.05 × 10^-8 C(
b) Tension in the stringT = mg / sin θT = (0.00190 kg × 9.81 m/s²) / sin 37.0°T = 3.38 × 10^-2 NThus, the charge on the ball is 1.05 × 10^-8 C and the tension in the string is 3.38 × 10^-2 N.
About Electric fieldThe electric field is an electric force that affects the space around electric charges. The cause of the electric field is the presence of positive and negative electric charges. The electric field can be described as lines of force or field lines. The electric field has units of N/C or read Newton/coulomb.The electric field is a vector quantity that exists at every point in space and is visualized as an arrow. So, every electrically charged object will produce an electric field or area that is affected by the electric force.
Learn More About Electric field at https://brainly.com/question/19878202
#SPJ11
In 4.4 years, the planet Zolton moves halfway around its orbit, a circle of radius 3.50×10
11
m centered on Helioz, its sun. (a) What is the average speed in this interval? km/s (b) What is the magnitude of the average velocity for this interval? km/s OHANPSE3 4.P.058. An audio compact disk (CD) player is rotating at an angular velocity of 3.6 radians per second when playing a track at 3.8 cm. (a) What is the linear speed at that radius? cm/s (b) What is the rotating rate in revolutions per minute? rev/min
Radius of the planet, r = 3.50×[tex]10^{11}[/tex] m Time taken by planet to move halfway around its orbit, t = 4.4 years = 4.4 x 365 x 24 x 60 x 60 s = 138384000 s
To find the average speed of planet, we can use the formula:
Average speed = Total distance travelled / Time taken
Total distance travelled by the planet when it moves halfway around its orbit is half of the circumference of its orbit.Hence,
Total distance travelled = πr= 3.14 x 3.50×[tex]10^{11}[/tex] = 1.099×[tex]10^{12}[/tex] m
Therefore,
Average speed = Total distance travelled / Time taken= 1.099×[tex]10^{12}[/tex] / 138384000= 7939.9 m/s ≈ 7.94 km/s
Therefore,
the average speed of planet Zolton in this interval is 7.94 km/s.
(b) To find the magnitude of the average velocity of planet, we need to find the displacement of the planet from its initial position to final position during the given time interval.Halfway around its orbit means, the planet comes back to its initial position.
To know more about halfway visit:
https://brainly.com/question/28740448
#SPJ11
what is the weight of air for the entire atmosphere
The weight of the entire atmosphere is approximately 5.2 x 10^18 kilograms.
The weight of the entire atmosphere can be calculated by multiplying the average density of the Earth's atmosphere by the total volume of the atmosphere.
The average density of the Earth's atmosphere at sea level is approximately 1.225 kilograms per cubic meter (kg/m³). The total volume of the atmosphere can be estimated using the mean radius of the Earth, which is about 6,371 kilometers (6,371,000 meters).
To calculate the weight of the atmosphere:
Weight = Density × Volume
Volume = (4/3) × π × (radius)^3
Weight = 1.225 kg/m³ × [(4/3) × π × (6,371,000 meters)^3]
Calculating this yields a weight of approximately 5.2 x 10^18 kilograms.
Therefore, the estimated weight of the entire atmosphere is approximately 5.2 x 10^18 kilograms.
To learn more about Density visit: https://brainly.com/question/1354972
#SPJ11
2- Prove that the molecular field of a dielectric material is Em = E +: P 38 where E is the macroscopic electric field.
The molecular field of a dielectric material, denoted as Em, can be expressed as Em = E +: P, where E is the macroscopic electric field and P is the polarization vector. This equation represents the sum of the external electric field and the electric field induced by the polarization of the material.
In the presence of an external electric field (E), dielectric materials exhibit polarization, where the alignment of molecular dipoles creates an internal electric field (Em) within the material. The molecular field (Em) can be defined as the sum of the external field (E) and the field induced by the polarization (P) of the material, expressed as Em = E +: P.
The polarization vector (P) represents the dipole moment per unit volume and is related to the electric susceptibility (χe) of the material through the equation P = χe * E. The electric susceptibility characterizes the material's response to an applied electric field.
When the material is non-polarizable (χe = 0), there is no induced polarization, and Em reduces to E. In this case, the molecular field is equal to the macroscopic electric field. However, in polarizable dielectric materials, the polarization induced by the external field contributes to the molecular field, resulting in Em being greater than E.
Hence, the expression Em = E +: P captures the relationship between the macroscopic electric field (E) and the molecular field (Em), accounting for the polarization effects in dielectric materials.
Learn more about macroscopic here:
https://brainly.com/question/30706053
#SPJ11
An object is located 20.8 cm in front of a convex mirror, the image being 8.00 cm behind the mirror. A second object, twice as tall as the first one, is placed in front of the mirror, but at a different location. The image of this second object has the same height as the other image. How far in front of the mirror is the second object located? Number Units
The second object is located at a distance of 41.67cm from the mirror.
We apply the mirror formula, and the equation for magnification as per required to arrive at the answer.
The mirror formula goes as follows.
1/f = 1/u + 1/v
The two forms of magnification go as follows.
m = h(i) / h(o) = -v/u
Where v = image distance from the pole
u = object distance from the pole
First, we apply the mirror formula to get the focal length.
1/f = -1/20.8 + 1/-8
1/f = 1/-0.173
f = -5.78 cm
Now, by applying the magnification formula for both objects of the same image height.
For object 1:
h(i) / h(o) = -8/-20.8 = 0.384
For object 2:
h'(i) / h'(o) = h(i) / 2h(o) = 0.384/2 = 0.192
But h'(i) / h'(o) = -v'/u'
=> -v/u = 0.192
u = -8/0.192 (v' = v)
u' = 41.66cm
Therefore, the second object is located at a distance of about 41.67 cm from the mirror.
For more using the mirror formula,
brainly.com/question/14343950
#SPJ4
Pushing down on a unicycle pedal with \( 272 \mathrm{~N} \) of force, the pedal fixed at \( 0.19 \mathrm{~m} \) from the center of the gear moves through \( 40 .^{\circ} \) of angle. What is the work
The work done to effect the motion of the unicycle pedal is approximately 92.363 newton-meters.
To find the work done, we'll use the formula W = F d cos(theta). The force applied on the pedal is given as 272 N.
The displacement, d, is the distance moved by the pedal, which is 0.19 m. The angle between the force and displacement vectors, theta, is 40 degrees. Now we can calculate the work done:
W = F d cos(theta)
= 272 N *0.19 m *cos(40 degrees)
First, we need to convert the angle from degrees to radians, as cosine expects the angle to be in radians. Converting 40 degrees to radians gives approximately 0.698 radians. Continuing the calculation:
W = 272 N * 0.19 m * cos(0.698 radians)
= 92.363 N * m
Therefore, the work done to effect the motion of the unicycle pedal is approximately 92.363 newton-meters.
Learn more about force here:
https://brainly.com/question/30507236
#SPJ11
A single slit diffraction pattern is projected on an image screen perpendicular to the light rays coming through the stit. The wavelength of the light is 600 * 10 m. The first dark fringe is located on the image screen at an angle equal to 30from the line from the slit to the center of the central bright fringe. The width W of the slit is (circle one answer) ? Oa: 600 x 10% O b. 1200 x 10 OC 300 x 10 m O d. 2400 x 10° 10 PM O e 1800 x
The width of the slit is 1200 * [tex]10^-^9 m[/tex], which corresponds to option (b) in the choices provided. To determine the width of the slit in a single-slit diffraction pattern, we are given the wavelength of the light, the angle of the first dark fringe, and the angle from the slit to the center of the central bright fringe.
The formula for the angle of the dark fringe in a single-slit diffraction pattern is given by the equation sinθ = mλ/W, where θ is the angle of the dark fringe, m is the order of the fringe (in this case, m = 1 for the first dark fringe), λ is the wavelength of the light, and W is the width of the slit.
Given that the angle of the first dark fringe is 30 degrees and the wavelength is 600 * 10^-9 m, we can rearrange the formula to solve for the width of the slit:
W = mλ / sinθ
W = (1)(600 * [tex]10^-^9 m[/tex]) / sin(30 degrees)
W = 600 *[tex]10^-^9 m[/tex] / 0.5
W = 1200 * [tex]10^-^9[/tex] m
Therefore, the width of the slit is 1200 * [tex]10^-^9[/tex]m, which corresponds to option (b) in the choices provided.
Learn more about diffraction here:
brainly.com/question/12290582
#SPJ11
What is the gravitational field a distance d above the center of
a uniformly-dense disk
of radius R?
Please, write the answer neatly.
The gravitational field at a distance d above the center of a uniformly-dense disk of radius R can be calculated using the following formula:
g = (2 * G * σ * R² * d) / (R² + d²)^(3/2)
Where:
g is the gravitational field strength,
G is the gravitational constant (approximately 6.67430 × 10^(-11) m³ kg^(-1) s^(-2)),
σ is the surface mass density (mass per unit area) of the disk.
Please note that the surface mass density, σ, should be provided for a more specific calculation.
To know more about density please click :-
brainly.com/question/29775886
#SPJ11
You are sitting on the periphery of your spaceship, fighting off aliens. The spaceship is a
kind of a flying saucer – a cylinder with radius 20 meters and mass 1500 kg (together
with you). You shoot a single shell from your blaster in a tangential direction. The mass
of the shell is 1 kg, the speed is 5000 m/s. Find the angular velocity that the spaceship
will acquire after the shot.
To find the angular velocity that the spaceship will acquire after the shot, we can apply the principle of conservation of angular momentum. The initial angular momentum of the spaceship and you together is equal to the final angular momentum of the spaceship after the shot.
The angular momentum is given by the equation:
�
=
�
⋅
�
L=I⋅ω
Where:
L is the angular momentum,
I is the moment of inertia, and
ω (omega) is the angular velocity.
The moment of inertia of a solid cylinder about its axis of rotation is given by:
�
=
1
2
⋅
�
⋅
�
2
I=
2
1
⋅m⋅r
2
Where:
m is the mass of the object, and
r is the radius of the object.
In this case, the initial angular momentum is zero because the spaceship is initially at rest. After the shot, the angular momentum is:
�
final
=
�
spaceship
⋅
�
spaceship
+
�
shell
⋅
�
shell
L
final
=I
spaceship
⋅ω
spaceship
+I
shell
⋅ω
shell
Since the shell is shot tangentially, its angular velocity (
�
shell
ω
shell
) is equal to its linear velocity (
�
shell
v
shell
) divided by the radius (
�
r) of the spaceship.
�
shell
=
�
shell
�
ω
shell
=
r
v
shell
Plugging in the values, we can calculate the angular velocity of the spaceship:
�
final
=
(
1
2
⋅
�
spaceship
⋅
�
spaceship
2
)
⋅
�
spaceship
+
(
1
2
⋅
�
shell
⋅
�
shell
2
)
⋅
�
shell
�
spaceship
L
final
=(
2
1
⋅m
spaceship
⋅r
spaceship
2
)⋅ω
spaceship
+(
2
1
⋅m
shell
⋅r
shell
2
)⋅
r
spaceship
v
shell
Now we can solve for
�
spaceship
ω
spaceship
:
�
spaceship
=
�
final
−
(
1
2
⋅
�
shell
⋅
�
shell
2
)
⋅
�
shell
�
spaceship
1
2
⋅
�
spaceship
⋅
�
spaceship
2
ω
spaceship
=
2
1
⋅m
spaceship
⋅r
spaceship
2
L
final
−(
2
1
⋅m
shell
⋅r
shell
2
)⋅
r
spaceship
v
shell
Plugging in the given values of the mass, radius, and velocity, we can calculate the angular velocity.
To learn more about principle of conservation of angular momentum, Click here:
https://brainly.com/question/29490733
#SPJ11
According to the second law of thermodynamics, energy cannot be created or destroyed. Therefore, both matter and energy are continuously recycled through ecosystems True False
According to the second law of thermodynamics, energy cannot be created or destroyed. Therefore, both matter and energy are continuously recycled through ecosystems is True.
The second law of thermodynamics states that in any energy transfer or transformation, the total amount of energy in a closed system remains constant, but the quality of the energy decreases. This means that energy cannot be created or destroyed, but it can change from one form to another (such as from chemical energy to heat energy) or be transferred between objects.
In ecosystems, matter and energy are constantly cycling and being recycled. Organisms obtain energy from food sources, convert it into various forms of energy for their own use, and release it back into the environment. Nutrients and other forms of matter are also recycled as they are taken up by organisms, transformed, and returned to the environment through processes like decomposition.
So, both matter and energy are continuously recycled through ecosystems in accordance with the second law of thermodynamics.
To know more about second law of thermodynamics here
https://brainly.com/question/7206767
#SPJ4
Problem 24 The diagram shows the right foot of a 120 lb person standing on "tip-toe." The tension in the Achilles tendon T makes an angle a 57 with the horizonta A. Neglecting the weight of foot (WF0), compute the tension in the Achilles tendon and the magnitude and direction of the reaction force R at the distal end of the tibia. Human). Assume the foot's center of gravity is 2 inches below the end of the tibia. weight? Is it reasonable to neglect the foot weight for this analysis? B. Repeat part A, but include the weight of foot (using the anthropometric data for the Standard C. Compare the answers in parts A and B. What percent error is introduced by omitting the foot 2" 2.3" 3.8 Answers A. B. C. T = 118.2 lb, R = 171.7 lb, ? = 68.0° down from the horizontal T-e 118.2 lb, R-: 170.11b, ?-67.8-down from the horizontal percent error in R is 0.95%, percent error in o is 0.32%
the percent error in the magnitude of the reaction force is 0.94%, and the percent error in the direction is 0.29%.
A. According to the provided answer, neglecting the weight of the foot, the tension in the Achilles tendon (T) is 118.2 lb, and the magnitude of the reaction force (R) at the distal end of the tibia is 171.7 lb. The direction of the reaction force is 68.0° down from the horizontal.
B. Including the weight of the foot, the tension in the Achilles tendon (T) is 118.2 lb, and the magnitude of the reaction force (R) at the distal end of the tibia is 170.1 lb. The direction of the reaction force is 67.8° down from the horizontal.
C. To calculate the percent error, we can use the formula: percent error = (|experimental value - accepted value| / accepted value) × 100%.
For the tension in the Achilles tendon:
Percent error = (|118.2 lb - 118.2 lb| / 118.2 lb) × 100% = 0%.
For the magnitude of the reaction force:
Percent error = (|171.7 lb - 170.1 lb| / 170.1 lb) × 100% = 0.94%.
For the direction of the reaction force:
Percent error = (|68.0° - 67.8°| / 67.8°) × 100% = 0.29%.
To know more about magnitude please click :-
brainly.com/question/31022175
#SPJ11
how is the phenotype of a quantitative trait expressed?
The phenotype of a quantitative trait is expressed through continuous variation. Quantitative traits are those that exhibit continuous variation over a range of phenotypes.
These traits are usually influenced by multiple genes, as well as the environment, resulting in a range of values rather than distinct categories. The phenotype of a quantitative trait can be expressed in various ways, including the mean, variance, and standard deviation. The mean of a quantitative trait refers to the average value of the trait among a group of individuals. The variance of a quantitative trait refers to the variation in the trait values within a population. Finally, the standard deviation of a quantitative trait refers to the degree of variation among individuals in the population. These measures are commonly used to describe the expression of quantitative traits and are used to study the underlying genetic and environmental factors that contribute to their expression.
To know more about phenotype please refer to:
https://brainly.com/question/22117
#SPJ11
A certain crystal is cut so that the rows of atoms on its surface are separated by a distance of 0.5 nm. A beam of electrons is accelerated through a potential difference of 150 V and is incident normally on the surface. If all possible diffraction orders could be observed, at what angles (relative to the incident beam) would the diffracted beams be found?
The diffracted beams would be found at angles corresponding to the diffraction orders given by the equation: sinθ = nλ/d, where θ is the angle of diffraction, n is the order of diffraction, λ is the wavelength of the electrons, and d is the distance between the rows of atoms on the crystal surface.
In this case, the wavelength of the electrons can be determined using the de Broglie wavelength equation: λ = h/p, where h is the Planck's constant and p is the momentum of the electrons.
To calculate the momentum of the electrons, we can use the equation: p = √(2meV), where me is the mass of an electron and V is the potential difference through which the electrons are accelerated.
Substituting the value of λ in the diffraction equation, we have: sinθ = n(h/p)/d.
By substituting the value of p, we can simplify the equation to: sinθ = n(h/√(2meV))/d.
Now, we can calculate the values of sinθ for different diffraction orders (n = 1, 2, 3, ...) by substituting the given values of h, me, V, and d.
Finally, by taking the inverse sine (sin⁻¹) of each value of sinθ, we can determine the corresponding angles θ at which the diffracted beams would be found.
Learn more about diffraction from the following link:
https://brainly.com/question/12290582
#SPJ11.
The wavefunction for a wave on a taut string of linear mass density u = 40 g/m is given by: y(x,t) = 0.25 sin(5rt - TX + O), where x and y are in meters and t is in y seconds. The energy associated with two wavelengths on the wire is: - O E = 1.85 J O E = 3.08 J O E = 1.23 J O E = 3.70 J O E = 2.47 J
The energy associated with two wavelengths on the wire is approximately 1.23 J.
The energy associated with a wave on a taut string can be calculated using the formula:
E = (1/2) muω[tex].^{2}[/tex][tex]A^{2}[/tex]
Where:
E is the energy of the wave
m is the linear mass density of the string
u is the angular frequency of the wave
A is the amplitude of the wave
In this case, the linear mass density (u) is given as 40 g/m, which can be converted to kg/m by dividing by 1000:
m = 40 g/m / 1000 = 0.04 kg/m
The angular frequency (ω) can be calculated using the formula:
ω = 2πf
Where f is the frequency of the wave. In this case, the frequency is given as:
f = 1 ÷ T = 1 / y seconds
The wave number (k) is given by:
k = 2π ÷ λ
Where λ is the wavelength of the wave. In this case, the wavelength (λ) is given by:
λ = 2π ÷ r
Where r is the constant in the wave function (5 in this case).
Now, let's calculate the energy associated with two wavelengths on the wire.
First, we need to find the frequency (f) and the wave number (k) using the given values:
f = 1 ÷ T = 1 ÷ y = 1 ÷ 2πr
k = 2π ÷ λ = 2π ÷ (2π÷r) = r
Now, we can calculate the angular frequency (ω) and the energy (E):
ω = 2πf = 2π ÷ (2πr) = 1÷r
E = (0.5) muω[tex].^{2} A^{2}[/tex] = (1/2) (0.04 kg/m) [tex]\frac{1}{r} ^{2} A^{2}[/tex]
Since we want to calculate the energy associated with two wavelengths, we can substitute the wavelength (λ) into the formula:
E = (0.5) (0.04 kg/m) [tex]\frac{1}{r} ^{2} A^{2}[/tex] = (0.5) (0.04 kg/m)[tex]\frac{1}{\frac{2\pi }{r} ^{2}} A^{2}[/tex]
Simplifying the equation:
E = (0.02 kg/m) [tex]\frac{4\pi ^{2} }{r^{2} }[/tex] [tex]A^{2}[/tex]
Now, we need to find the value of r from the wave function:
y(x, t) = 0.25 sin(5rt - TX + O)
Comparing this with the general form of the wave function:
y(x, t) = Asin(kx - ωt + φ)
We can see that r = 5r, so:
r = 5
Substituting this value back into the equation for energy:
E = (0.02 kg/m) [tex]\frac{4\pi ^{2} }{5^{2} }[/tex] [tex]A^{2}[/tex]
E ≈ 1.23 J
Learn more about wavelengths here:
https://brainly.com/question/30611426
#SPJ11
A particle in an experimental apparatus has a velocity given by v=k
s
, where v is in millimeters per second, the position s is millimeters, and the constant k=0.28 mm
1/2
s
−1
. If the particle has a velocity v
0
=3 mm/s at t=0, determine the particle position, velocity, and acceleration as functions of time. To check your work, evalutate the time t, the position s, and the acceleration a of the particle when the velocity reaches 15 mm/s. Answers:
t=
s=
a=
s
mm
mm/s
2
The correct answer is t = 0.0141 s,s = 3084.5 mm,a = -2.936 mm/s^2 (approx). Velocity of particle, v = k s,where k = 0.28 mm^(1/2)s^(-1), s is the position in mm. Particle velocity, v0 = 3 mm/s, at t = 0.
: We know that, v = k s. Differentiating both sides with respect to time, we get,dv/dt = k ds/dt.
Here, ds/dt = v/kSo, dv/dt = k v/k = k^(1/2)v.
Differentiating again with respect to time, we get,d^2s/dt^2 = d/dt(k^(1/2)v)d^2s/dt^2 = k^(1/2)dv/dt.
Therefore, d^2s/dt^2 = k^(1/2)×k^(1/2)v = k v = k(k s) = k^2 s.
Here, we have the differential equation of acceleration as,d^2s/dt^2 = k^2 s.
Now, the standard form of this equation is given by,d^2y/dx^2 + k^2 y = 0.
Comparing the above equations, we have,y = s, x = t.
Therefore, the solution of the above differential equation is given by,s = Asin(kt) + Bcos(kt), where A and B are constants.
Substituting the initial condition, v0 = 3 mm/s at t = 0.
We have, v = k s = k[Asin(kt) + Bcos(kt)]At t = 0, v = 3 mm/sSo, 3 = k[Bcos(0)] = Bk.
Therefore, B = 3/kAlso, v = k s = k[Asin(kt) + Bcos(kt)]v = kAsin(kt) + 3, at t = 0⇒ 3 = kA⇒ A = 3/k.
Therefore, v = k[3/k sin(kt) + 3/k cos(kt)] = 3sin(kt) + 3cos(kt) = 3 sin(kt + π/4).
Thus, position of the particle as a function of time is,s = 3/k sin(kt) + 3/k cos(kt) = 3/k sin(kt + π/4).
Differentiating s w.r.t. t, we get,ds/dt = 3k/k cos(kt) - 3k/k sin(kt)ds/dt = 3k/k(cos(kt) - sin(kt))ds/dt = 3(cos(kt) - sin(kt)).
Differentiating again w.r.t. t, we get,d^2s/dt^2 = -3k sin(kt) - 3k cos(kt)d^2s/dt^2 = -3k(sin(kt) + cos(kt))d^2s/dt^2 = -3[cos(kt + π/2)]d^2s/dt^2 = -3sin(kt).
Therefore, acceleration as a function of time is given by a = -3sin(kt).
Now, given, velocity of particle, v = k s,where k = 0.28 mm^(1/2)s^(-1), s is the position in mm.
To determine the time t, when the velocity reaches 15 mm/s, we have,15 = k s(t)At t = 0, v = 3 mm/s.
Let, at time t, the velocity is 15 mm/s, then we have,15 = k s(t) => 15 = 0.28 s(t)^(1/2) => s(t) = (15/0.28)^2s(t) = 3084.5 mm.
Now, we have s(t) = 3/k sin(kt) + 3/k cos(kt)At t = t0, when the velocity reaches 15 mm/s, we have s(t0) = 3084.5 mm and, v(t0) = 15 mm/s.
From the equation, v = k[3/k sin(kt) + 3/k cos(kt)], we get,15 = 0.28[3/k sin(kt0) + 3/k cos(kt0)] => 53.57 = sin(kt0) + cos(kt0).
From the above equation, we can solve for t0 by substituting sin(kt0) = 53.57 - cos(kt0) and taking cos(kt0) common,53.57 - cos(kt0) = cos(kt0) (tan(kt0) + 1).
On solving the above equation, we get,t0 = 0.0141 s.
Thus, time t = t0 = 0.0141 s, position s = s(t0) = 3084.5 mm, acceleration a = -3sin(kt0) = -2.936 mm/s^2 (approx).
Hence, the required answers are,t = 0.0141 s,s = 3084.5 mm,a = -2.936 mm/s^2 (approx).
Learn more about acceleration here ;
https://brainly.com/question/2303856
#SPJ11