Sphere On Incline (Rotational Kinematics and Energy)

• Pat2666
In summary, a solid sphere of uniform density with a mass of 4.2 kg and a radius of 0.28 m starts from rest and rolls without slipping down a 22° incline for a distance of 2.6 m. The fraction of kinetic energy that is translational is 0.71, the translational kinetic energy at the bottom is 28.66 J, and the translational speed at the bottom is 3.69 m/s. To find the magnitude of the frictional force, the linear acceleration can be found and used with Newton's 2nd law for translation.
Pat2666
Okay, another problem from me!

http://img187.imageshack.us/img187/69/32aa2d25kf9.jpg

A solid sphere of uniform density starts from rest and rolls without slipping a distance of d = 2.6 m down a q = 22° incline. The sphere has a mass M = 4.2 kg and a radius R = 0.28 m.

--------------------------------------------------------------------------------
a) Of the total kinetic energy of the sphere, what fraction is translational?
KEtran/KEtotal = *
0.71 OK

--------------------------------------------------------------------------------
b) What is the translational kinetic energy of the sphere when it reaches the bottom of the incline?
KEtran = J *
28.66 OK

--------------------------------------------------------------------------------
c) What is the translational speed of the sphere as it reaches the bottom of the ramp?
v = m/s *
3.69 OK

--------------------------------------------------------------------------------
d) What is the magnitude of the frictional force on the sphere?
|f| = N

HELP: The frictional force provides the torque needed to give an angular acceleration. Therefore, first find the angular acceleration, then apply the rotational equivalent of Newton's 2nd Law (torque = I*a).
HELP: Since the sphere rolls without slipping, the angular acceleration is related to the translational acceleration, which can be found from kinematic relations.

Alrighty, so I managed to figre out all of the problem except this last bit. I thought I knew what to do after reading over the two HELP bits provided, but clearly I'm doing something wrong.

Since I know the Vt at the bottom of the ramp, and it being zero at the top of the ramp, I thought I could use that to solve for a (as you will see below). I then plug that into the modified equation for torque, but get the wrong answer. I don't fully understand how torque is equivilant to the force of friction, as that's what the HELP seems to say, but I'd also like to understand what I'm doing.

My Work :

http://img187.imageshack.us/img187/5945/workje4.jpg

Last edited by a moderator:
Pat2666 said:
Since I know the Vt at the bottom of the ramp, and it being zero at the top of the ramp, I thought I could use that to solve for a (as you will see below).
Sure.
I then plug that into the modified equation for torque, but get the wrong answer.
What's the "modified" equation for torque? T = I*alpha. How does force relate to torque?
I don't fully understand how torque is equivilant to the force of friction, as that's what the HELP seems to say, but I'd also like to understand what I'm doing.
The friction is the only force that exerts a torque on the sphere (about its center of mass). But the force and the torque it produces are two different things. (They even have different units.)

Since you've found the linear acceleration, you can also just apply Newton's 2nd law for translation to find the friction force.

Ahhh, so I solved torque, but that's isn't to force, it's the product of the force and radius.

Thanks for the help :)

I knew something had to be wrong with my answer since I was pretty sure torque =/= the frictional force.

1. What is rotational kinematics?

Rotational kinematics is the study of the motion of objects that are rotating or moving in a circular path. It involves concepts such as angular velocity, angular acceleration, and angular displacement.

2. How does rotational energy relate to a sphere on an incline?

In the context of a sphere on an incline, rotational energy refers to the energy associated with the sphere's rotation as it rolls down the incline. This energy is a combination of its rotational kinetic energy and its potential energy due to its position on the incline.

3. What is the equation for rotational kinetic energy?

The equation for rotational kinetic energy is KE = 1/2 * I * ω^2, where I is the moment of inertia and ω is the angular velocity. This equation shows that rotational kinetic energy depends on the object's mass, shape, and how fast it is rotating.

4. How does the angle of incline affect the motion of a sphere?

The angle of incline affects the motion of a sphere by determining the sphere's speed and acceleration as it rolls down the incline. A steeper incline will result in a higher speed and acceleration, while a shallower incline will result in a slower speed and acceleration.

5. What is the relationship between rotational and translational motion in a sphere on an incline?

In a sphere on an incline, rotational and translational motion are related through the concept of rolling without slipping. This means that the sphere's rotational and translational velocities are directly proportional, and its rotational and translational accelerations are equal in magnitude but opposite in direction.

Replies
19
Views
3K
Replies
10
Views
1K
Replies
8
Views
2K
Replies
97
Views
4K
Replies
35
Views
3K
Replies
5
Views
1K
Replies
3
Views
1K
Replies
7
Views
5K
Replies
5
Views
1K
Replies
6
Views
3K