Sphere rolling up an incline then back down

In summary: If you want to, why not figure the time it takes to go down the incline using the distance x that you already calculated? (You'd better get the same...or nearly the same answer, since that's all you're given)That sounds like a good idea. Thanks.
  • #1
shizzle
21
0
Need Help Here!

A sphere or mass, m and radius r rolls along a horizontal surface with a constant velocity, Vi approaches an incline with (angle theta). ie bottom angle :smile: If it rolls without slipping,
a) what is the maximum distance,x it will travel on the incline?
b) If it begins to roll back down, find the time it takes to get to horizontal surface.
c)What will be its final velocity. ie. at time it gets to horizontal surface.

My solution:

I want to write energy equations and use conservation of energy to solve it.

K.E. = 1/2M(Vi^2) + 1/2I(w^2) where I = 2/5 r^2
P.E = mgh(max) - kNx where k is coeficient of friction and N is normal force

using trig, h(max) = xsin(theta) so x = h(max)/sin(theta)
so
P.E = mgh (max) - [kNh(max)] /sin(theta) ; N = mgcos(theta)
P.E = mgh(max) - [kmgcos(theta)h(max)]/ sin(theta)

h(max) is the vertical distance traveled ie. less than h and
x is distance traveled on incline i.e less than d

Are my equations right? and if so,the way to go now is to substitute w= v/r, set K.E = P.E and solve for x?
 
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  • #2
shizzle said:
I want to write energy equations and use conservation of energy to solve it.
Good. That's how to solve part a).

K.E. = 1/2M(Vi^2) + 1/2I(w^2) where I = 2/5 r^2
I = 2/5 M r^2
P.E = mgh(max) - kNx where k is coeficient of friction and N is normal force
The friction does no work, since it rolls without slipping.

using trig, h(max) = xsin(theta) so x = h(max)/sin(theta)
Right.
Are my equations right?
Correct your expression for PE and you're good to go.
and if so,the way to go now is to substitute w= v/r, set K.E = P.E and solve for x?
Right.
 
  • #3
Hey Doc Al,
Thanks. I think i figured out the 3rd part.

I first found the acceleration.
since w = v/r ; v = wr
a = w(dot -on top of it--hehe) r
w(dot) = a / r

I w(dot) = Fr where F is friction
ma = mg sin (theta) -F

If you plug expression for F into it
a = 5/7g sin(theta)

Now, knowing acceleration, i just separate variables and integrate to find velocity

and i got it to be 5/7gsin(theta) t

If this is right, i'll need to figure out the time it takes to get down. do i just make t the subject? that seems trivial...any help?
 
  • #4
shizzle said:
If you plug expression for F into it
a = 5/7g sin(theta)
Looks good to me.

Now, knowing acceleration, i just separate variables and integrate to find velocity

and i got it to be 5/7gsin(theta) t
Right.

If this is right, i'll need to figure out the time it takes to get down. do i just make t the subject? that seems trivial...any help?
It's as easy as you think it is. :smile: [itex]V = 5/7 g sin\theta t[/itex], since you know the final speed [itex]V = V_i[/itex], just solve for t.
 
  • #5
okay, but when i solve for t, my expression will involve v not vi. Should i then make vi the subject of my x expression from a) and plug vi into my t? (the vi expression will involve v meaning my final t expression will involve v)

I'm also not really sure why you say v = vi at the end. Is it because the ball goes back to rolling on the horizontal surface? Thanks. I'm almost there:smile:
 
  • #6
shizzle said:
I'm also not really sure why you say v = vi at the end. Is it because the ball goes back to rolling on the horizontal surface?
Yes. Since the ball started up the incline with a linear speed of Vi, that's what it will end up with when it rolls back down.
 
  • #7
so t = 7v/5gsin (theta) but v = vi so
t = 7vi/5sin(theta)?

It just seems weird that we're suddenly replacing the velocity with which it hits the ground with its initial velocity (even though i understand that vf = vi)

Is there something I'm missing? or am i just thinking too hard?
 
  • #8
shizzle said:
so t = 7v/5gsin (theta) but v = vi so
t = 7vi/5sin(theta)?
Right (but don't leave out the g).

It just seems weird that we're suddenly replacing the velocity with which it hits the ground with its initial velocity (even though i understand that vf = vi)

Is there something I'm missing? or am i just thinking too hard?
It may seem weird, but vi is the only information you are given, so your answer had better be in terms of it!

If you want to, why not figure the time it takes to go down the incline using the distance x that you already calculated? (You'd better get the same answer!)
 

1. How does the mass of the sphere affect its motion when rolling up an incline?

The mass of the sphere has a direct effect on its motion when rolling up an incline. A heavier sphere will require more force to move up the incline, while a lighter sphere will require less force. This is due to the relationship between mass and inertia, where a greater mass results in a greater resistance to change in motion.

2. What role does friction play in the motion of a sphere rolling up an incline?

Friction plays a crucial role in the motion of a sphere rolling up an incline. Without friction, the sphere would continue to roll up the incline indefinitely. However, friction acts in the opposite direction of motion, slowing down the sphere and eventually bringing it to a stop. This is why a sphere will eventually roll back down the incline, as friction overcomes the force of gravity pulling it down.

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

The angle of the incline has a significant impact on the motion of a rolling sphere. As the angle increases, the force of gravity pulling the sphere down the incline also increases. This results in a faster acceleration of the sphere, causing it to roll up the incline at a faster speed. On the other hand, a smaller angle will result in a slower acceleration and a slower speed for the rolling sphere.

4. Can a sphere ever roll up an incline and then stop at the top?

No, a sphere cannot roll up an incline and then stop at the top without external forces acting upon it. This is because of the principle of conservation of energy, which states that energy cannot be created or destroyed, only transferred or transformed. When a sphere is rolling up an incline, it gains potential energy due to its increased height. This potential energy is then converted into kinetic energy as the sphere rolls back down the incline.

5. How does the surface of the incline affect the motion of a rolling sphere?

The surface of the incline can have a significant impact on the motion of a rolling sphere. A rough surface will provide more friction, resulting in a slower rolling speed and a shorter distance traveled by the sphere. On the other hand, a smooth surface will provide less friction, allowing the sphere to roll at a faster speed and cover a greater distance. The material of the incline can also affect the amount of friction, with a rougher material providing more friction than a smoother material.

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