Common Rim Speed

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109.Common Rim Speed

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Two pulleys are connected by a non-slipping belt. Pulley 1 has radius

3R

, pulley 2 has radius

R

, and pulley 1 is driven at a constant angular speed

\omega

. The pulleys spin about fixed parallel axes and the belt is taut. Three points are marked:

A

is fixed on the rim of pulley 1,

B

is fixed on the rim of pulley 2, and

C

is painted on the belt and is currently located on the straight segment between the two pulleys (not yet in contact with either pulley). Which of the following correctly orders the magnitudes of the lab-frame accelerations

a_A

a_B

, and

a_C

$Side view of two pulleys on fixed parallel axes. The left pulley (pulley 1) is large with radius $3R$; the right pulley (pulley 2) is small with radius $R$. A taut belt wraps around both pulleys, with visible straight segments along the top and bottom connecting them. Mark point $A$ on the rim of the large pulley at its topmost position. Mark point $B$ on the rim of the small pulley at its topmost position. Mark point $C$ on the straight upper segment of the belt, roughly midway between the two pulleys. Show a curved arrow on pulley 1 indicating its angular velocity $\omega$.$

a_C < a_A < a_B

a_C < a_B < a_A

a_C < a_A = a_B

a_A < a_B = a_C

a_B = a_C < a_A

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