Finding Maximum Acceleration on a Box on a Cart with Friction

[amcavoy]

This is a link to a picture of my problem:

http://img57.imageshack.us/img57/9382/cart5rh.jpg

It asks what tension on the rope would give the cart a maximum acceleration. The surface the cart is on is frictionless, while the cart itself is not (the surface the box is on).

To solve this, I figured that pulling the rope with a force of u_sN would give the maximum acceleration of the cart. However, this seems too easy; I think I may have not completely answered the question.

|T| = u_sN = u_smg

Is my reasoning correct? Thanks.

 

[Doc Al]

You really didn't provide your reasoning, just your answer. (No, it's not correct.)

Instead, think this way: What's the only force directly applied to the cart? What's the maximum acceleration acceleration that that force can give the cart? Use that to find the required tension in the rope.

 

[Doc Al]

What contact force is exerted on the cart?

 

[amcavoy]

You really didn't provide your reasoning, just your answer. (No, it's not correct.)
Instead, think this way: What's the only force directly applied to the cart? What's the maximum acceleration acceleration that that force can give the cart? Use that to find the required tension in the rope.

Ok this is what I came up with:

$$F_M=\mu_sN\implies a_M=\frac{\mu_sN}{M}$$

$$T=\left(M+m\right)a=\frac{\left(M+m\right)\mu_sN}{M};\quad N=mg$$

Is this correct?

Thanks again.

 

[Doc Al]

Now you've got it.

 

[amcavoy]

Thanks Doc Al, I appreciate it.