Rotational Kinetic Energy AP Physics C Mechanics FRQ – Exam Style Questions etc.
Rotational Kinetic Energy AP Physics C Mechanics FRQ
Unit 6: Energy and Momentum of Rotating Systems
Weightage : 10-15%
QUESTION
A wind turbine includes a three-blade system that rotates about an axis through the end of each blade, as shown in
Figure 1. Each blade has a length L and mass M , with a center of mass located at a distance $\frac{L}{3}$ from the axis of
rotation, as shown in Figure 2.
(a) Derive an expression for the rotational inertia of the three-blade system. Express your answer in terms of M,
L, and physical constants, as appropriate. The rotational inertia of each blade about an axis through its center
of mass is given by the equation $I_{cm}=\frac{1}{18}ML^{2}$.
(b) While the wind blows, the three-blade system operates at a constant angular speed $ω_{0} = 2.6 rad/s$. The length
of one blade is L = 36 m. The numerical value of the rotational inertia of the system is $I_{sys}=6.7\times 10^{6}kg.m^{2}$.
Calculate the time T it takes the outer edge of a single blade to complete one revolution.
(c) When the wind stops blowing, the angular speed of the system decreases. The angular speed ω of the system
while slowing down is given as a function of time t by the equation $\omega =\omega _{0}e^{-\beta _{0^{t}}}$, where $β_{0}$ is a constant with
appropriate units, as shown on the graph in Figure 3.
i. Calculate the amount of energy dissipated from t = 0, when the wind stops blowing, until the system comes to rest.
ii. Derive an expression for the net torque exerted on the system as a function of t as the system slows down.
Express your answers in terms of $β_{0}, \omega _{0}, M, L, and I_{sys}$ and physical constants, as appropriate.
iii. Derive an expression for the angular displacement of the system Δθ as a function of t. Express your
answer in terms of $β_{0}, \omega _{0}, M, L, and I_{sys}$ and physical constants, as appropriate.
The three-blade system is now replaced with a second three-blade system identical to the first, except that the
second three-blade system slows down according to the equation $\omega = \omega _{0}e^{-\beta t}$, where w0 = 2.6 rad/s and $ β >β_{0}$.
The original angular speed function is shown as a dashed line in Figure 4.
(d) On the graph in Figure 4, sketch the angular speed of the second three-blade system as a function of time t.
▶️Answer/Explanation
Ans:-
(a) For stating the parallel axis theorem
(b) For calculating the correct answer with correct units ( 2.4 s)
(c)(i) For indicating that the total initial rotational kinetic energy is dissipated
(c)(ii) For using Newton’s second law in rotational form
(c)(iii) For attempting to integrate the expression for angular speed
(d) For drawing a continuous curve showing an exponential decay
Question
A solid disk of unknown mass and known radius R is used as a pulley in a lab experiment, as shown above. A small block of mass m is attached to a string, the other end of which is attached to the pulley and wrapped around it several times. The block of mass m is released from rest and takes a time t to fall the distance D to the floor.
a. Calculate the linear acceleration a of the falling block in terms of the given quantities.
b. The time t is measured for various heights D and the data are recorded in the following table.
| D(m) | t(s) |
| 0.5 | 0.68 |
| 1 | 1.02 |
| 1.5 | 1.19 |
| 2 | 1.38 |
i. What quantities should be graphed in order to best determine the acceleration of the block? Explain your reasoning.
ii. On the grid below, plot the quantities determined in b. i., label the axes, and draw the best-fit line to the data.
iii. Use your graph to calculate the magnitude of the acceleration.
c. Calculate the rotational inertia of the pulley in terms of m, R, a, and fundamental constants.
d. The value of acceleration found in b.iii, along with numerical values for the given quantities and your answer to c., can be used to determine the rotational inertia of the pulley. The pulley is removed from its support and its rotational inertia is found to be greater than this value. Give one explanation for this discrepancy.
Answer/Explanation
Ans:
a. x = v0t + ½ at2
x = D and v0 = 0 so D = ½ at2 and a = 2D/t2
b. i. graph D vs. t2 (as an example)
iii. a = 2(slope) = 2.04 m/s2
c. Στ = TR = Iα and α = a/R so I = TR2/a
ΣF = mg – T = ma so T = m(g – a)
I = m(g – a)R2/a = mR2 ((g/a) – 1)
d. The string was wrapped around the pulley several times, causing the effective radius at which the torque acted to be larger than the radius of the pulley used in the calculation.
The string slipped on the pulley, allowing the block to accelerate faster than it would have otherwise, resulting in a smaller experimental moment of inertia. Friction is not a correct answer, since the presence of friction would make the experimental value of the moment of inertia too large
Question
The moment of inertia of a uniform solid sphere (mass M, radius R) about a diameter is 2MR²/5. The sphere is placed on an inclined plane (angle θ) as shown above and released from rest.
a. Determine the minimum coefficient of friction µ between the sphere and plane with which the sphere will roll down the incline without slipping
b. If µ were zero, would the speed of the sphere at the bottom be greater, smaller, or the same as in part a.? Explain your answer.
Answer/Explanation
Ans:
a. Torque provided by friction; at minimum µ, Ff = µFN = µMg cos θ
τ = FfR = Iα = (2/5)MR2 (a/R); Ff= (2/5)Ma = µMg cos θ giving a = (5/2)µg cos θ
ΣF = Ma; Mg sin θ – µMg cos θ = Ma = (5/2)µMg cos θ giving µ = (2/7) tan θ
b. Energy at the bottom is the same in both cases, however with µ = 0, there is no torque and no energy in rotation, which leaves more (all) energy in translation and velocity is higher