Source: High school physics (Chinese)
Problem Sets:
Problem
As shown in Figure, a uniform disk with radius $R=15$ cm and mass $M=3.0$ kg is mounted on a fixed horizontal axle through its center. A light string is wrapped around its circumference with no slipping. A constant downward force $F=5.0$ N is applied to the string.
- Find the angular acceleration of the disk.
- Find the tangential acceleration of the disk's edge.
P0627-problem-1
[Q1] $\alpha = 22.2$ rad/s² [Q2] $a_t = 3.33$ m/s²
The torque on the disk is produced by the applied force $F$ at a radius $R$, so $\tau = FR$. The rotational inertia of a uniform disk about its center is $I = \frac{1}{2}MR^2$. Using Newton's second law for rotation, $\tau = I\alpha$:
$$FR = \left(\frac{1}{2}MR^2\right)\alpha$$Solving for the angular acceleration $\alpha$:
$$\alpha = \frac{FR}{\frac{1}{2}MR^2} = \frac{2F}{MR}$$The tangential acceleration $a_t$ at the edge is related to the angular acceleration by $a_t = \alpha R$.
$$a_t = \left(\frac{2F}{MR}\right)R = \frac{2F}{M}$$