Torques of equal magnitude are applied to a hollow cylinder and a solid sphere, both having the same mass and radius. The cylinder is free to rotate about its standard axis of symmetry, and the sphere is free to rotate about an axis passing through its centre. Which of the two will acquire a greater angular speed after a given time?
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Let angular accelerations produced by hollow cylinder and solid sphere are a1 and a2 respectively.
We know,
Torque = moment of inertia × angular acceleration
So,
Torque of hollow cylinder = torque of solid sphere
I1 × a1 = I2 × a2
Moment of inertia of hollow cylinder = mR²
Moment of inertia of solid sphere = (2/5)mR²
mR² × a1 = (2/5)mR² × a2
a2 = 2.5a1
After a given, time w1 and w2 be the angular speed of hollow cylinder and solid sphere respectively.
W1 = Wo + a1t
W2 = Wo + 2.5a1t
hence, w2 > w1
e.g solid sphere will acquire gretar angular speed after a given time .
We know,
Torque = moment of inertia × angular acceleration
So,
Torque of hollow cylinder = torque of solid sphere
I1 × a1 = I2 × a2
Moment of inertia of hollow cylinder = mR²
Moment of inertia of solid sphere = (2/5)mR²
mR² × a1 = (2/5)mR² × a2
a2 = 2.5a1
After a given, time w1 and w2 be the angular speed of hollow cylinder and solid sphere respectively.
W1 = Wo + a1t
W2 = Wo + 2.5a1t
hence, w2 > w1
e.g solid sphere will acquire gretar angular speed after a given time .
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