Rotational Equilibrium and Newton’s First Law in Rotational Form AP Physics C Mechanics FRQ – Exam Style Questions etc.
Rotational Equilibrium and Newton’s First Law in Rotational Form AP Physics C Mechanics FRQ
Unit 5: Torque and Rotational Dynamics
Weightage : 10-15%
Question
A uniform disk of radius R and mass \(m_{d}\) is attached to a vertical pole by a horizontal axle that passes through the center of the disk. Friction between the axle and the disk is negligible. A lump of clay of mass \(m_{c}\) is attached to the edge of the disk at Point A. The size of the lump of clay is small compared with the radius of the disk. A horizontal string is connected from the pole to the edge of the disk at Point A. The string makes an angle \(\Theta\) with the line between Point A and the axle, as shown in Figure 1.
(a) On the following representation of the clay-disk system, draw and label the external forces (not components) exerted on the system. Each force must be represented by a distinct arrow that starts on, and points away from, the point at which the force is exerted on the system.
(b) Derive an expression for the tension \(F_{T}\) in the string when the clay is at Point A, as shown in Figure 1, in terms of R, \(m_{d}\), \(m_{c}\), \(\Theta\), and physical constants, as appropriate.
(c) The string remains connected to the edge of the disk at Point A. The clay is moved to Point B, which is horizontally in line with the axle, as shown in Figure 2. How does the new tension \(F_{T},_{new}\) compare with tension \(F_{T}\) from part (b)? Justify your reasoning.
(d) A nonuniform disk is now attached to the axle. The lump of clay is attached to the disk at Point B, as shown in Figure 3. The clay has mass \(m_{c}\) = 0.60 kg and the disk has a radius R = 0.30 m. The mass density of the disk varies radially and can be modeled by \(\rho \left ( r \right ) =\beta r\), where r is the radial distance from the axle and \(\beta \) = \(4.0kg/m^{3}\) .
i. Calculate the rotational inertia of the disk about the axle.
ii. The string connecting the disk to the pole is cut. Calculate the magnitude of the initial angular acceleration of the clay-disk system.
▶️Answer/Explanation
3(a)Example Responses
Scoring Note: Examples of appropriate labels for the gravitational force include \(F_{G}\), \(F_{g }\), \(F_{grav}\), W , mg , Mg , “grav force,” “ F Earth on disk,” “ F on disk by Earth,” \(F_{Earth\\\ on \\\ Disk}\), \(F_{E},_{Disk}\) , and \(F_{Disk},_{E}\) . The labels G or g are not appropriate labels for the gravitational force.
Scoring Note: Examples of appropriate labels for the normal force from the axle include \(F_{N}\), \(F_{axle }\), N , “normal force,” and “axle force.”
Scoring Note: Examples of appropriate labels for the tension force include \(F_{t}\), \(F_{T}\), T , \(F_{string}\), and “Force from string.”
3(b)Example Responses
Scoring Note: A maximum of three points can be earned if the trigonometric functions ( sin and cos ) are reversed for both torque expressions.
3(c)Example Responses
The torque caused by the weight of the clay at Point B is greater than when the clay is at Point A because the component of the weight that is perpendicular to the lever arm is larger. To maintain equilibrium, the net torque on the system is still zero, therefore the tension \(F_{T},_{new}\) must be greater.
3(d)(i)Example Solution
3(d)(ii)Example Solution
