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A belt moves over two pulleys $A$ and $B$ as shown in figure. The pulleys are mounted on two fixed horizontal axles. Radii of the pulleys $A$ and $B$ are $50 c m$ and $80 c m$ respectively. Pulley $A$ is driven at constant angular acceleration of $0.8 \frac{r a d}{s^{2}}$ until pulley $B$ acquires an angular velocity of $10 \frac{r a d}{s}$ . The belt does not slide on either of the pulleys. Find the acceleration of a point $C$ on the belt and angular acceleration of the pulley $B$ .

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$a_{C} = 0.4 m / s^{2}; α_{B} = 0.5 \frac{r a d}{s^{2}}$

$a_{C} = 0.4 m / s^{2}; α_{B} = 0.8 \frac{r a d}{s^{2}}$

$a_{C} = 0.2 m / s^{2}; α_{B} = 0.8 \frac{r a d}{s^{2}}$

$a_{C} = 0.2 m / s^{2}; α_{B} = 0.5 \frac{r a d}{s^{2}}$

answer is A.

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Detailed Solution

Since the belt does not slide on the pulleys, magnitudes of velocity and acceleration of any point on the belt are same as that of any point on the periphery of either of the two pulleys.

$\vec{a_{T}} = \vec{α} \times \vec{r} a_{C} = α_{A} r_{A} = α_{B} r_{B}$

Substituting $r_{A} = 0.5 m, r_{B} = 0.8 m$ and $α_{A} = 0.8 \frac{r a d}{s^{2}}$ ,

we have $a_{C} = 0.4 m / s^{2}$ and $α_{B} = \frac{a_{C}}{r_{B}} = \frac{α_{A} r_{A}}{r_{B}} = 0.5 \frac{r a d}{s^{2}}$

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