Sphere rolling down a ramp

In summary, the problem involves a 2kg hollow spherical shell rolling without slipping down a 38 degree slope. The task is to find the acceleration, friction force, and minimum coefficient of friction required to prevent slipping. The equations used are mgh and E=1/2mv^2=1/2Iw^2, and the gravity provides a torque of mg*cos(38). Hints are given to help solve the problem, such as writing an equation for the energy of the object and determining the torque that causes rotation. It is also important to consider whether acceleration will be constant and use additional equations of motion if necessary.
  • #1
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Homework Statement



A hollow sphereical shell rolls without slipping down a 38 degree slope, mass 2kg, find the acceleration the friction force and the minimum coefficent of friction needed to prevent slipping

Homework Equations



mgh, E=1/2mv^2=1/2Iw^2

The Attempt at a Solution



i don't really know where to start, i know moment of inertia will affect how fast this will accelerate due to gravity but I am not given it so i don't know to solve it, hints would be great please!
 
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  • #2
the gravity provides the ball with a torque which is mg*cos(38)
when the friction is big enough to cancel out the turning effect caused by the torque
 
  • #3
wwj said:
the gravity provides the ball with a torque which is mg*cos(38)
when the friction is big enough to cancel out the turning effect caused by the torque

How can this be a torque when the units are wrong?
 
  • #4
Time for some hints. First of all write an equation energy of the object when it has moved an arbitrary distance (D) along the slope.

Next, determine the torque that causes the ball to rotate.

Think about whether acceleration will be a constant. Based on your decision, more equations of motion might come to mind.
 
  • #5




To solve this problem, we can use the concept of torque and the equations of motion for a rolling object. The moment of inertia for a hollow spherical shell is given by I = 2/3mr^2, where m is the mass of the sphere and r is its radius. We can use this to calculate the angular acceleration of the sphere as it rolls down the ramp.

The torque on the sphere is due to the force of friction, which is equal to the coefficient of friction multiplied by the normal force (mgcosθ). The normal force is equal to the weight of the sphere, mg, multiplied by the cosine of the angle of the slope (θ). So the torque can be written as μmgcos^2θ, where μ is the coefficient of friction.

Using the equation for torque, τ = Iα, we can set this equal to the torque due to friction and solve for the angular acceleration, α. This will give us the acceleration of the sphere down the ramp.

To prevent slipping, the friction force must be equal to or greater than the maximum static friction force, which is equal to μsN, where μs is the coefficient of static friction and N is the normal force. So we can set the torque due to friction equal to μsmgcos^2θ and solve for the minimum coefficient of static friction needed to prevent slipping.

Finally, to find the acceleration of the sphere down the ramp, we can use the equation of motion for a rolling object, a = αr, where r is the radius of the sphere. This will give us the linear acceleration of the sphere.

I hope this helps to guide you in solving this problem. Remember to carefully consider all the forces acting on the sphere and use the appropriate equations to solve for the desired values.
 

1. How does the height of the ramp affect the speed of the sphere?

The height of the ramp directly affects the speed of the sphere. As the height of the ramp increases, the potential energy of the sphere also increases. This means that the sphere will have more energy to convert into kinetic energy as it rolls down the ramp, resulting in a higher speed.

2. What factors influence the acceleration of the sphere rolling down a ramp?

The acceleration of the sphere depends on several factors, including the angle of the ramp, the mass of the sphere, and the surface of the ramp. A steeper ramp will result in a higher acceleration, while a smoother surface will decrease friction and increase acceleration. The mass of the sphere also plays a role, with heavier spheres experiencing less acceleration due to their greater inertia.

3. How does the radius of the sphere affect its speed on the ramp?

The radius of the sphere does not have a significant effect on its speed on the ramp. As long as the surface of the ramp is smooth and the sphere is rolling without slipping, the radius will not change the speed. However, a larger radius may result in a longer time to reach the bottom of the ramp due to the increased distance traveled.

4. What is the relationship between the length of the ramp and the distance the sphere travels?

The length of the ramp is directly proportional to the distance the sphere travels. This means that as the length of the ramp increases, the distance the sphere travels also increases. This is due to the fact that the longer the ramp, the more time the sphere has to accelerate and cover a greater distance.

5. How does the mass of the sphere affect its speed on the ramp?

The mass of the sphere has a direct impact on its speed on the ramp. Heavier spheres have more inertia and require more force to accelerate, resulting in a lower speed on the ramp. On the other hand, lighter spheres will accelerate faster and have a higher speed on the ramp.

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