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

The mass of the block m is:

`m=M(sin(\theta)+\mu_(s)cos(\theta))`

Step-by-step 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!