What is the upward acceleration of M2 in this system with given forces and mass?

[ganondorf29]

Homework Statement

Mass m1=31.7 kg is on a horizontal surface, connected to mass m2= 5.30 kg by a light string as shown. The pulley has negligible mass and no friction. A force of 216.9 N acts on m1 at an angle of 29.3deg. The coefficient of kinetic friction between m1 and the surface is 0.221. Determine the upward acceleration of m2.

Homework Equations

F=ma
Fn = mg-Fsin(theta)
Fcos(theta)-(uk)(Fn) = ma

The Attempt at a Solution

First I found the normal force of M1. Fn = mg-Fsin(theta). I found Fn to be 204.513 N. Next, I found the acceleration of M1 by using [Fcos(theta)-(uk)(Fn)] / m = a. I got ax to be 4.54 m/s^2. I assumed that the acceleration would be the same for M2 but its not.

 

[Doc Al]

You forgot to consider the string tension acting on the block. (The accelerations of M1 and M2 must be the same--they are connected by a string.)

 

[baba89]

To find the acceleration of M2, we can use the fact that the tension in the string connecting M1 and M2 is the same for both masses. This means that the force pulling M2 upward is also 216.9 N. We can then use F=ma to find the acceleration of M2, which is 216.9 N / 5.30 kg = 40.93 m/s^2. This means that M2 is accelerating upward at 40.93 m/s^2. However, this is not the final answer since we need to take into account the friction between M2 and the surface.

The friction force acting on M2 is equal to the coefficient of kinetic friction (uk) multiplied by the normal force (Fn). In this case, the normal force acting on M2 is equal to its weight, which is mg = 5.30 kg * 9.8 m/s^2 = 51.94 N. So the friction force is 0.221 * 51.94 N = 11.48 N.

Now, we can use F=ma again, taking into account the friction force, to find the acceleration of M2. The equation would be F - ukFn = ma, where F is the force pulling M2 upward (216.9 N) and ukFn is the friction force (11.48 N). Solving for a, we get a = (216.9 N - 11.48 N) / 5.30 kg = 40.48 m/s^2.

So the final answer is that M2 is accelerating upward at 40.48 m/s^2. This is slightly less than the acceleration of M1, which makes sense because M2 has the additional force of friction acting against it.