First slide

Rotational motion

Question

Two forces F and 2F are applied on a rod of length l and mass m, as shown. The angular acceleration of the rod is

Moderate

A

$\frac{8 F}{m l} (- \hat{k})$
B

$\frac{4 F}{m l} (\hat{k})$
C

$\frac{4 F}{m l} (- \hat{k})$
D

$\frac{8 F}{m l} (\hat{k})$

Solution

Force equation $F - 2 F = m a \Rightarrow a = - \frac{F}{m}$
Hence, $\vec{a} = - \frac{F}{m} \hat{i}$
Torque equation about center of mass
$\begin{array}{l} F \cdot \frac{l}{2} + (\frac{2}{3} l - \frac{l}{2}) = I_{C} α \\ F \cdot \frac{l}{2} + F \frac{l}{6} = \frac{m l^{2}}{12} \cdot α \\ \frac{2}{3} F l = \frac{m l^{2}}{12} \cdot α \Rightarrow α = \frac{8 F}{m l} \\ Hence, \vec{α} = - \frac{8 F}{m l} \hat{k} \end{array}$

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