Question

A square metal plate with mass M = 1. 20 [kg] and side length 0.180 [m] is pivoted

about an axis through point O at its center and perpendicular to the plate. Three

forces of magnitude F1 = 18. 0 [N], F2 = 26. 0 [N], and F3 = 14. 0 [N] act on the

plate as shown.

a. What is the net torque on the square plate due to these forces?

b. What is the angular acceleration of the disk? The moment of inertia for a plate of length

a and width b rotating about its center is Icom = 1⁄12 M(a^2 + b^2).

c. If the plate is initially at rest, what is the angular velocity of the particle after 3. 00 [s]?

Answer 1

a. $\tau=\tau_2+\tau_3-\tau_1$

$\tau_1=r_1F_1\sin\alpha_1$

$\tau_2=r_2F_2\sin\alpha_2$

$\tau_3=r_3F_3\sin\alpha_3$

$r_1=r_2=r_3=r$

$r=\sqrt{(L/2)^2+(L/2)^2}=L/\sqrt2=0.18/\sqrt2=0.127\ (m)$

$\tau_1=0.127\cdot 18\cdot\sin135°=1.62\ (N\cdot m)$

$\tau_2=0.127\cdot 26\cdot\sin135°=2.33\ (N\cdot m)$

$\tau_3=0.127\cdot 14\cdot\sin90°=1.78\ (N\cdot m)$

$\tau=2.33+1.78-1.62=2.49\ (N\cdot m)$ . Answer

b. $\tau=I\epsilon\to \epsilon=\tau/I$

$I=\frac{1}{12}m(a^2+b^2)=\frac{1}{12}\cdot1.2\cdot(0.18^2+0.18^2)=0.00648\ (kg\cdot m^2)$

$\epsilon=2.49/0.00648=384\ (rad/s^2)$ . Answer

c. $\omega=\omega_0+\epsilon t=0+384\cdot3=1152\ (rad/s)$ . Answer