Cylinder rolling down a wedge - wedge should not slip

[Saitama]

Homework Statement

The height of the slope shown in the figure is 30 cm, and its base is 40 cm. A solid cylinder of uniform density, which has the same mass as the slope, rolls down along the slope without slipping. What is the least value of the coefficient of friction between the inclined plane and the horizontal ground if the slope does not slip?

Homework Equations

The Attempt at a Solution

I understand that this is a simple problem but I cannot reach the specified answer though I did a similar problem recently.

The forces acting on the wedge are the normal reaction from cylinder (N), normal reaction from ground (N'), weight (mg) and the frictional force (f).

Balancing forces on the wedge in the vertical direction:
$$N'=mg+N\cos\theta$$
where ##\theta## is the angle made by the slope with the horizontal.
Since the wedge doesn't slip,
$$f \geq N\sin\theta \Rightarrow \mu N' \geq N\sin\theta$$
Using ##N=mg\cos\theta## and solving the equations, I don't get the right answer.

Any help is appreciated. Thanks!

 

[TSny]

The forces acting on the wedge are the normal reaction from cylinder (N), normal reaction from ground (N'), weight (mg) and the frictional force (f).

How many friction forces act on the wedge?

 

[voko]

The cylinder has to spin up as it rolls down without slipping, which means there is some torque on it, which is provided by a force. By Newton's third law, the wedge will experience the opposite force.

 

[Saitama]

How many friction forces act on the wedge?

The cylinder has to spin up as it rolls down without slipping, which means there is some torque on it, which is provided by a force. By Newton's third law, the wedge will experience the opposite force.

Silly me, thanks a lot both of you! :)

 

[Nickie]

Your approach to this problem is correct. It seems that there may be an error in your calculations. Here is a step-by-step solution:

1. Draw a free body diagram of the wedge to identify the forces acting on it.
2. Apply Newton's second law in the horizontal and vertical directions separately to find the normal reactions N and N'.
3. Use the condition for no slipping, which states that the friction force f must be greater than or equal to the component of the weight of the cylinder along the slope, which is mgsinθ.
4. Substitute the expression for N found in step 2 into the inequality from step 3.
5. Simplify the inequality and solve for the coefficient of friction μ.

If you follow these steps, you should arrive at the correct answer. Remember to double check your calculations to ensure accuracy. Keep up the good work!