Question

A block with mass M is placed on an inclined plane with slope angle q and is connected to a second hanging block with mass m by a cord passing over a small, frictionless pully. The coefficient of static friction μs and the coefficient of kinetic friction is μk. For what range values of of m will the blocks remain at rest if they are released from rest.

Answer 1

Answer:

The mass of the block m is:

$m=M(sin(\theta)+\mu_{s}cos(\theta))$

Explanation:

Let's analyze the block by parts

For the block M

$T-W_{x}-f_{f}=0$ (1)

Where:

• T is the tension
• W(x) is the component of the weight in the x-direction
• F(f) is the friction force

$T-Mgsin(\theta)-\mu_{s}N=0$  

$T-Mgsin(\theta)-\mu_{s}Mgcos(\theta)=0$  

For the block m

$T-W=0$

$T=mg$ (2)

Now, let's combines equation (1) and (2):

$mg-Mgsin(\theta)-\mu_{s}Mgcos(\theta)=0$  

Finally, let's solve it for block m.

$mg-Mg(sin(\theta)+\mu_{s}cos(\theta))=0$

$m=M(sin(\theta)+\mu_{s}cos(\theta))$

I hope it helps you!