Theta_max = 53.8° Acceleration of Hoop on Inclined Ramp

[cmcc3119]

A circular hoop of mass m, radius r, and infinitesimal thickness rolls without slipping down a ramp inclined at an angle theta with the horizontal. View Figure

PART A. What is the acceleration of the center of the hoop?

Express the acceleration in terms of physical constants and all or some of the given quantities: m,r and theta.
= (g/2)*sin(theta)

IS THIS CORRECT? It is the answer I got but I could not see any reason to have r and m in the final equation. I am scared to enter it as the computer might say it is wrong like this!



PART B. What is the minimum coefficient of friction needed for the hoop to roll without slipping?

Express the minimum coefficient of friction in terms of all or some of the given quantities: m, r, and theta
= 1/2*tan(theta)

IS THIS CORRECT? Again as above this is the answer I got but I could not see any reason to have r and m in the final equation. I am scared to enter it as the computer might say it is wrong like this!



PART C: Imagine that the above hoop is a tire. The coefficient of static friction between rubber and concrete is typically at least 0.9. What is the maximum angle theta_max you could ride down without worrying about skidding?
Express your answer numerically, in degrees, to two significant figures.

I am confused as it is asking for the maximum angle. Does that mean I still use the minimum coefficient. Can someone please start me off as I think my answers to the other two are correct but this one I am hesitant on.

Thank you lovely intelligent people!

 

[Doc Al]

Parts A and B are correct. But next time show your work, don't just give an answer. Part C is just the inverse of part B. For any given angle, you know the minimum coefficient required; just work it in reverse.

 

[Moetasim]

RESPONSE:

Hello,

I can provide a response to your questions. Let's start with Part A. The acceleration of the center of the hoop can be expressed as:

a = (g/2)*sin(theta)

This is indeed the correct answer and it does not require the mass or radius of the hoop. This is because the acceleration of an object rolling without slipping down an inclined ramp is only dependent on the angle of the ramp and the acceleration due to gravity (g). The radius and mass of the hoop do not affect the acceleration in this case.

Moving on to Part B, the minimum coefficient of friction needed for the hoop to roll without slipping can be expressed as:

μ = 1/2*tan(theta)

Again, this is the correct answer and it does not require the mass or radius of the hoop. This is because the minimum coefficient of friction needed for an object to roll without slipping is dependent on the angle of the ramp and the gravitational acceleration (g). The radius and mass of the hoop do not affect this minimum coefficient.

For Part C, we are dealing with a different scenario where the hoop is now a tire. The maximum angle theta_max that you could ride down without worrying about skidding can be calculated using the coefficient of static friction between rubber and concrete (μ = 0.9). We can rearrange the equation from Part B to solve for theta:

μ = 1/2*tan(theta)

tan(theta) = 2μ

theta = arctan(2μ)

Plugging in μ = 0.9, we get:

theta_max = arctan(2*0.9) = 54.5°

Therefore, the maximum angle theta_max you could ride down without worrying about skidding is approximately 54.5°.

I hope this helps clarify any confusion you had. Keep up the good work!