The L-shaped object in the figure below consists of three masses (m 1 = 9.06 kg,

Question

The L-shaped object in the figure below consists of three masses (m1 = 9.06 kg, m2 = 1.02 kg and m3 = 2.52 kg) connected by light rods of length L1 = 1.11 m and L2 = 2.07 m.

a) What torque must be applied to this object to give it an angular acceleration of 1.33 rad/s2 if it is rotated about the x axis?

b) What torque must be applied to this object to give it an angular acceleration of 1.33 rad/s2 if it is rotated about the y axis?

c) What torque must be applied to this object to give it an angular acceleration of 1.33 rad/s2 if it is rotated about the z axis (which is through the origin and perpendicular to the page)?

Explanation / Answer

The moment of inertia about some axis is given by:
I = m1(r1)2 + m2(r2)2 + m3(r3)2
where r1, r2, and r3 are the distances from the axis to m1, m2, and m3 respectively.

Also, torque = (moment of inertia)(angular acceleration).

Rotating about the x-axis:
Since m2 and m3 are on the x-axis, r2 = r3 = 0, and r1 = L1.
So:
I = m1(L1)2 = (9.06 kg)(1.11 m)2 = 11.16kg*m2
and
T = Ia = (11.16kg*m2)(1.33 rad/s2) = 14.84 Nm

Rotating about the y-axis:
Since m1 and m2 are on the x-axis, r1 = r2 = 0, and r3 = L2.
So:
I = m3(L2)2 = (2.52 kg)(2.07 m)2 = 10.79 kg*m^2
and
T = Ia = ( 10.79 kg*m2)(1.33 rad/s2) = 14.35 Nm

Rotating about the z-axis (which is through the origin and perpendicular to the page):
Since m2 is on the x-axis, r2 = 0, r1 = L1, and r3 = L2.
So:
I = m1(L1)2 + m3(L2)2 = (9.06 kg)(1.11 m)2 + (2.52 kg)(2.07 m)2 = 22.95 kg*m2
and
T = Ia = (22.95 kg*m2)(1.33 rad/s2) = 30.52 Nm

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