Q3). A square frame is made from four thin sticks of mass m and length L. a) Sho
ID: 1293266 • Letter: Q
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Q3). A square frame is made from four thin sticks of mass m and length L. a) Show that the moment of inertia of the frame about an axis through its centre C and perpendicular to the plane of the frame is Ic = (4/3)mL^2. Use this result to show that the moment of inertia about an axis through the point O and perpendicular to the plane of the frame is Io = (10/3) mL^2. b) The frame is attached to a pivot at O about which it can rotate freely in a vertical plane. What is the initial angular acceleration of the frame after being released from the position shown? c) What is the angular velocity of the frame when the diagonal OE is vertical? d) What is the velocity of the point D at this instant of time?Explanation / Answer
(a) m L^2 /12 moment of inertia of stick about center
m L^2 / 2 + m (L/2)^2 = m L^2 /3 inertia of 1 stick about C
I = 4/3 m L^2 for all four sticks about C
[(L/2)^2 + (L/2)^2]^1/2 = (L/2)^1/2 distance from O to C
I = 4/3 m L^2 + 4 m L^2 / 2 = 10/3 m L^2 about O
(b) torque = 4 m g L 2^1/2 / 2 where 2^1/2 L /2 is distance from O to C
alpha = torque / Io = [2 m g L 2^1/2 ] / (10/3 m L^2)
alpha = 3 * 2^1/2 g / (5 L)
(c) PE = 4 m g L /2 = 2 m g L initial potential energy
KE = 1/2 Io w^2 = 5/3 m L^2 * w^2 final KE
w = 30^1/2 / 5 * (g / L)1/2 angular velocity when OE vertical
(d) w * L = V velocity of point D
V = 30^1/2 / 5 * (g L)^1/2
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