A flywheel of moment of inertia 0.32 kgm^2 is rotated steadily at 120rad/s by a 50 W electric motor. (i)find the kinetic energy and angular momentum of the flywheel. (ii) calculate the value of frictional couple opposing the rotation. (iii)find the time taken for the wheel to come to rest after the motor has been switched off.
(i) First compute the moment of inertia, I, with the formula for a disc or cyliner. The kinetic energy is (1/2)I w^2 and the angular momentum is I w.
(ii) The electric power driving the motion is T*w, where T is the torque. Set that equal to the 50W electrical power input and solve for T. The electrical motor torque in steady state operation equals the frictional torque. The words torque and couple are interchangeable.
(iii) During the slowdown time, t:
T*t = initial angular momentum = I w.
You know T from (ii) and I w from (i). . Solve for the time, t.
(i) Ah, the flywheel is having quite a spin! To find the kinetic energy, we can use the formula K.E. = (1/2) I ω^2, where I is the moment of inertia and ω is the angular speed.
K.E. = (1/2) * 0.32 kgm^2 * (120 rad/s)^2
K.E. = 2304 J (jokes)
For the angular momentum, we can use the formula L = I ω, where L is the angular momentum.
L = 0.32 kgm^2 * 120 rad/s
L = 38.4 kgm^2/s (keep spinning, flywheel!)
(ii) The frictional couple opposing the rotation is equal to the rate at which work is done against friction. Since the work done is equal to the power, P, we can use P = F * ω, where F is the frictional force and ω is the angular speed.
Frictional couple (C) = P/ ω
C = 50 W / 120 rad/s
C = 0.42 Nm (a little resistance never hurt anyone, right?)
(iii) To find the time taken for the wheel to come to rest, we can use the equation ω = ω_0 - αt, where ω_0 is the initial angular speed, ω is the final angular speed (zero in this case), α is the angular acceleration, and t is the time taken.
Since the acceleration is constant, we can use the equation α = τ/I, where τ is the torque and I is the moment of inertia.
α = -τ/I (negative sign indicates deceleration)
α = -C/I (using the frictional couple from part (ii))
α = -0.42 Nm / 0.32 kgm^2
α = -1.3125 rad/s^2
ω = ω_0 - αt
0 = 120 rad/s - (-1.3125 rad/s^2) * t
t = 120 rad/s / (1.3125 rad/s^2)
t ≈ 91.47 seconds (Time to say goodbye, flywheel!)
Please note that these calculations assume no other external forces are acting on the flywheel. If there are any other forces, the results may vary.
To calculate the kinetic energy of the flywheel, we can use the formula:
Kinetic Energy (KE) = (1/2) * moment of inertia * angular velocity^2
Given:
Moment of inertia (I) = 0.32 kgm^2
Angular velocity (ω) = 120 rad/s
Plugging in the values, we get:
KE = (1/2) * 0.32 * (120)^2
= 2304 J
Therefore, the kinetic energy of the flywheel is 2304 J.
To calculate the angular momentum, we can use the formula:
Angular Momentum (L) = moment of inertia * angular velocity
Given:
Moment of inertia (I) = 0.32 kgm^2
Angular velocity (ω) = 120 rad/s
Plugging in the values, we get:
L = 0.32 * 120
= 38.4 kgm^2/s
Therefore, the angular momentum of the flywheel is 38.4 kgm^2/s.
To calculate the value of the frictional couple opposing the rotation, we need to use the following formula:
Frictional Couple (C) = Power (P) / Angular velocity (ω)
Given:
Power (P) = 50 W
Angular velocity (ω) = 120 rad/s
Plugging in the values, we get:
C = 50 / 120
= 0.4167 Nm
Therefore, the value of the frictional couple opposing the rotation is approximately 0.4167 Nm.
To find the time taken for the flywheel to come to rest after the motor has been switched off, we need to use the formula:
Time (t) = Initial angular velocity (ω) / Angular deceleration (α)
Given:
Initial angular velocity (ω) = 120 rad/s
Final angular velocity = 0 (since the wheel comes to rest)
Moment of inertia (I) = 0.32 kgm^2
We can find the angular deceleration (α) using the equation:
Final angular velocity^2 = Initial angular velocity^2 - 2 * α * θ
Since the final angular velocity is 0 and the initial angular velocity is 120 rad/s, we can solve for α:
0 = (120)^2 - 2 * α * θ
α = (120)^2 / (2 * θ)
Using the given moment of inertia (I) = 0.32 kgm^2, we can calculate the angular deceleration (α) as follows:
α = (120)^2 / (2 * 0.32)
= 5400 rad/s^2
Finally, we can calculate the time taken using the formula:
t = (Final angular velocity - Initial angular velocity) / Angular deceleration
t = (0 - 120) / (-5400)
= 0.0222 s
Therefore, the time taken for the flywheel to come to rest after the motor has been switched off is approximately 0.0222 seconds.
To find the answers to these questions, we'll need to use the formulas and principles of rotational motion.
(i) Kinetic Energy of the flywheel:
The kinetic energy of a rotating object can be calculated using the formula:
KE = (1/2) I ω^2
Where:
KE is the kinetic energy,
I is the moment of inertia,
ω is the angular velocity.
Given that the moment of inertia of the flywheel is 0.32 kgm^2 and the angular velocity is 120 rad/s, we can substitute these values into the formula to find the kinetic energy.
KE = (1/2) * 0.32 * (120)^2
= 2304 J
Angular Momentum of the flywheel:
Angular momentum is given by the formula:
L = I ω
Where:
L is the angular momentum,
I is the moment of inertia,
ω is the angular velocity.
Using the given values, we can substitute them into the formula to find the angular momentum.
L = 0.32 * 120
= 38.4 kgm^2/s
(ii) Frictional Couple opposing the rotation:
The power provided by the electric motor is 50 W. Since power is the rate at which work is done, and work is equal to the product of force and distance, we can calculate the force opposing the rotation.
P = τω
Where:
P is the power,
τ is the frictional couple (torque),
ω is the angular velocity.
Given that the power is 50 W and the angular velocity is 120 rad/s, we can rearrange the equation to solve for the frictional couple.
τ = P/ω
= 50/120
= 0.417 Nm
(iii) Time taken for the wheel to come to rest:
When the motor is switched off, the only force acting on the flywheel is the frictional couple. The angular acceleration can be determined by using the formula:
τ = I α
Where:
τ is the frictional couple (torque),
I is the moment of inertia,
α is the angular acceleration.
Since the motor is switched off, the final angular velocity will be zero (ω = 0). Rearranging the equation, we get:
α = τ / I
Substituting the values given, we have:
α = 0.417 / 0.32
= 1.303 rad/s^2
To find the time taken for the wheel to come to rest, we can use the equation:
ω = ω0 + αt
Since the final angular velocity is 0 (ω = 0), we can rearrange the equation to solve for the time (t):
t = -ω0 / α
= -120 / 1.303
≈ -92.1 s
Since time cannot be negative, the time taken for the wheel to come to rest after the motor has been switched off is approximately 92.1 seconds.