Rigid Body Motion: Sphere on Inclined Plane

In summary, the sphere is moving on a inclined plane at a velocity Vo. The ball is moving with the sphere and is at a position X(t) as a function of time. The work done by friction is equal to -Ffr X(t).
  • #1
hellsnake
2
0

Homework Statement



A sphere of mass M and radius R it`s given a velocity Vo on the base of a inclined plane (theta being the angle) and friction coefficient mu (you may assume that static and kinetic friction are equal) Find the position of the ball as a function of time

Homework Equations





The Attempt at a Solution

 
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  • #2
Welcome to PF!

Hi hellsnake! Welcome to PF! :smile:

(have a theta: θ and a mu: µ :wink:)

(and btw, is the sphere skidding, or is it rolling without slipping?)

Use work done and conservation of energy …

what do you get? :smile:
 
  • #3
It doesn'st tell you if it's slipping or not you have to see that. By the way I forgot you have to study the two cases when µ≤ 2/7 tan θ and µ≥ 2/7 tan θ. Without the equal. And when I do conservation of energy and work done by friction I get this differential equation
(1/2)M.(Vcm)² + (1/2)Icm.w² + MgH - (1/2)MVo² = -Ffr X(t)
But Vcm= (d/dt)X(t) and H=X(t)sin θ . Then I get that

(1/2)M.(d/dt)X(t))² + (1/2)Icm.w² + MgX(t)sin θ - (1/2)MVo² = -Ffr X(t)

Where:
M= mass of the sphere
Vcm= It's the velocity of the center of the mass
Icm= It's the moment or inertia with an axis that pass through the center oh the sphere
w= It's the angular velocity
H= It's how high is the sphere measure from the floor
Vo= It's the initial velocity
Ffr= It's the force that the frcition does
X(t)= It'sthe position of the ball measure from the point where it start his motion. And I choose the X axis parallel to the motion

So if I read the equation I get that the energy on a arbitrary point of the sphere is equal to
(1/2)M.(Vcm)² + (1/2)Icm.w² + MgH. Now this minus the energy in the initial moment(the sphere doesn't have rotational energy here) (1/2)MVo² . All of this is equal to the work done by friction -Ffr X(t)

I think that I have to do another thing because this differential equation It's too hard for this course I think. However if you have the solution for this equation I appreciate it too

Thanks a lot in advance
 
  • #4
Hi hellsnake! :smile:

(have an omega: ω :wink:)

You can simplify it slightly by writing Icm in terms of M and R, and writing Ffr = µMg.

After that, you need a relation between V and ω … try the easier, rolling-without-slipping case first: V = Rω. :wink:

(and after that, you'll need to find the condition on µ for rolling … presumably it's going to be µ ≥ 2/7 tanθ ! :rolleyes:)
 

1. What is rigid body motion?

Rigid body motion refers to the movement of an object as a whole, without any deformation or change in shape. In other words, all points of the object move together in the same way.

2. What is the sphere on inclined plane problem?

The sphere on inclined plane problem is a classic physics problem that involves a spherical object rolling down an inclined plane. The goal is to calculate the acceleration, velocity, and position of the sphere at different points along the incline.

3. What are the key factors that influence the motion of the sphere on inclined plane?

The key factors that influence the motion of the sphere on inclined plane include the angle of the incline, the mass and radius of the sphere, the coefficient of friction between the sphere and the incline, and the force of gravity.

4. How is the motion of the sphere on inclined plane related to Newton's laws of motion?

The motion of the sphere on inclined plane can be described using Newton's laws of motion. The first law states that an object will remain at rest or in motion at a constant velocity unless acted upon by an external force. The second law relates the net force on an object to its acceleration, and the third law states that for every action, there is an equal and opposite reaction.

5. What are some real-life applications of the sphere on inclined plane problem?

The sphere on inclined plane problem has many real-life applications, such as understanding the motion of a rolling ball, predicting the trajectory of a bowling ball, and designing roller coasters and other amusement park rides. It also has practical applications in engineering, such as designing ramps, slopes, and conveyor belts.

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