# Find acceleration of a Block within a simply pulley system.

1. Dec 11, 2009

### cocoon

1. The problem statement, all variables and given/known data

The figure shows two blocks suspended by a cord over a pulley. The mass of black B is twice the mass of black A, $$m_b = 2m_a = 2m$$. The mass of the pulley is equal to the mass of black A, $$m_p = m_a = m$$, the radius of the pulley is $$R$$. The blocks are let free to move and the cord moves on the pulley without slipping or stretching. There is no friction in the pulley axle, and the cords are massless. The rotational inertia of the pulley is $$\frac{m_pR^2}{2}$$ about a perpendicular axis through the center. Find the magnitude of acceleration of the block B. Express your results in the simplest possible form in terms of any or all the following: m, g, R, and universal physical or mathematical constants.

2. Relevant equations

$$F_G = m * g$$ (force of gravity equals mass times gravitational acceleration)
$$\tau_net = I * \alpha$$ (torque equals moment of inertia times angular acceleration)

3. The attempt at a solution

$$\tau_{net} = I * \alpha = \frac{m * R^2}{2} * \alpha = \frac{m * R^3}{2} * a$$

furthermore,

$$\tau_{net} = F_{G on B} * R + F_{G on A} * R = 2 * m * g * R - m * g * R = m * g * R$$

so,

$$m * g * R = \frac{m * R^3}{2} * a$$

$$a = \frac{2 * g}{R^2} m/s^2$$

Is this correct? Is there an easier way?

Last edited: Dec 11, 2009
2. Dec 11, 2009

### cocoon

bump... picture of problem lol
http://img340.imageshack.us/img340/690/18485819.jpg [Broken]

Last edited by a moderator: May 4, 2017
3. Dec 11, 2009

### PhanthomJay

oops, slight error in your equation there $$a = \alpha R,$$ so $$\tau = MRa/2$$
no, that's not right, you've assumed the respective tension forces are equal to the respective weights, but the system is accelerating, so the tensions cannot equal the weights
No, this is incorrect, you need to write 3 equations with 3 unknowns, T_A, T_B, and a, so draw Free Body diagrams (FBD's ) of each block and the pulley to solve for them...... separte FBD's and application of Newton's laws to each are essential in these type problems. Trying to take shortcuts often results in incorerct methods and solutions.

4. Dec 11, 2009

### trust143_raj

this accn.. u solved is for t he pulley not for the block B
so use the principle of energy conversion to solve the problem then it will give the acceleration for block B..