The pulleys in figure (10-E6) are identical, each having a radius R and moment of inertia I. Find the acceleration of the block M.

Figure

Open in App

Solution

Free the body diagram of the system,

For block of mass M,
$M g - T_{1} = M a . . . (i)$
$(T_{1} - T_{2}) R = I α using, a = α r \Rightarrow (T_{1} - T_{2}) = I \frac{a}{R^{2}} . . . (i i) (For pully 1) Similarly, (T_{2} - T_{3}) = I \frac{a}{R^{2}} . . . (i i i) (For pully 2)$

For block of mass m,
$T_{3} - m g = m a . . . (i v) (For block m)$

Adding equations (ii) and (iii), we get:
$(T_{1} - T_{3}) = \frac{2 I a}{R^{2}} . . . (v)$
Adding equations (i) and (iv), we get:
$- m g + M g + (T_{3} - T_{1}) = M a + m a . . . (v i)$
Using equations (v) and (vi), we get:
$M g - m g = M a + m a + \frac{2 I a}{R^{2}} \Rightarrow a = \frac{(M - m) g}{(M + m + \frac{2 I}{R^{2}})}$

Suggest Corrections

Similar questions

R

I

M

r

I

The pulleys in figure (10 - E6) are identical, each having a radius R and moment of inertia I. Find the acceleration of the block M.

r

Join BYJU'S Learning Program