Newton’s Second Law in Rotational Form AP Physics C Mechanics FRQ – Exam Style Questions etc.
Newton’s Second Law in Rotational Form AP Physics C Mechanics FRQ
Unit 5: Torque and Rotational Dynamics
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