Minimum coefficient of friction to prevent slipping - torque

[ph123]

A hollow, spherical shell with mass 1.95 rolls without slipping down a slope angled at 30.0. Find the minimum coefficient of friction needed to prevent slipping.

I have already calculated the acceleration of the sphere and the magnitude of the friction force on the sphere in previous parts. I used torque and Newton's laws to find the acceleration and friction. They are:
a = 2.94 m/s^2
f = 3.82 N

I used the following equation for the sum of forces in the x-direction, which I oriented along the slope of the incline, with the positive direction down the incline. I didn't think I needed to use torque here, because I got the same answer that way as well.

(sum)Fx = mgsin(theta) - f = ma
= mgsin(theta) - (mu)mgsin(theta) = ma
mu = [a - gsin(theta)] / [-gsin(theta)]
mu = (-1.96 m/s^2) / (-4.9 m/s^2)
= 0.4

This coefficient of friction isn't right, but I'm sure I included all of the forces. Anyone know where I messed up? Thanks.

 

[AlephZero]

(sum)Fx = mgsin(theta) - f = ma
= mgsin(theta) - (mu)mgsin(theta) = ma

The friction force is mu times the normal force on the plane

f = (mu)mg COS(theta).

You don't need the rest of your equation. You already found f = 3.82N so just plug the numbers into that equation.

 

[m2003]

I would like to first commend you on your use of equations and principles such as torque and Newton's laws to calculate the acceleration and friction force in previous parts. Your approach is correct, and your calculation of the coefficient of friction is also correct. However, there seems to be a small error in the signs used in your equation for the sum of forces in the x-direction. The correct equation should be:

(sum)Fx = mgsin(theta) - f = ma
= mgsin(theta) - (mu)mgcos(theta) = ma
mu = [a + gsin(theta)] / [gcos(theta)]
mu = (1.96 m/s^2) / (4.25 m/s^2)
= 0.46

This slight change in the sign of the second term makes a significant difference in the value of the coefficient of friction. The correct value is 0.46, which is higher than the 0.4 you calculated.

In terms of understanding where the error occurred, it is important to remember that the friction force acts in the opposite direction to the motion, and in this case, the motion is down the incline. So, the friction force should be in the uphill direction, which is why the cosine function should be used instead of the sine function.

In conclusion, your use of equations and principles is correct, and the error lies in a small sign mistake. The correct minimum coefficient of friction needed to prevent slipping in this scenario is 0.46. Keep up the good work in your scientific calculations!