Uniform ball rolling without slipping problem

In summary, the problem involves a uniform ball rolling without slipping on a rough fixed sphere. The speed of the ball's center of mass is to be found in terms of the angle theta, and it is shown that the ball will leave the sphere when cos(theta) = 10/17. To find this, the energy principle is used and the normal force is set to zero. Newton's second law is also used to determine the forces acting on the ball and when the normal force becomes zero.
  • #1
pentazoid
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Homework Statement


A uniform ball of mass M and raduis a can roll without slipping on the rough outer surface of a fixed sphere of raduis b and centre O. Initially the ball is at rest at the highest point of the phere when it is slightly disturbed . Find the speed of the center the G of the ball in terms of the variable theta , the angle between the line OG and the upward vertical. [Assume planar motion]. Show that the ball will leave the sphere when cos (theta)=10/17


Homework Equations



linear momentum principle

M*dV/dt=dP/dt=F

The Attempt at a Solution



answer: v^2=(10/7)*g(a+b)(1-cos(theta)

since they give you v^2 and v^2 is associated with the kinetic energy of a particle, should I apply the Energy principle rather than the linear momentum principle.

I probably should break the x and y components of the ball with radius b and the hemisphere with to sin(theta) and cos(theta) components. the y- componet will contained the weight of the balls while the x-component will not.
 
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  • #2
Hi pentazoid! :smile:
pentazoid said:
should I apply the Energy principle rather than the linear momentum principle.

Yes! You must use energy to find the speed …

how could you apply linear momentum when there's no linear motion and there's gravity?

But you will need Newton's second law to find when the normal force is zero.
I probably should break the x and y components of the ball with radius b and the hemisphere with to sin(theta) and cos(theta) components. the y- componet will contained the weight of the balls while the x-component will not.

hmm … probably easier to use radial and tangential components …

the ball will lose contact when the normal (ie radial) force is zero. :wink:
 
  • #3
tiny-tim said:
Hi pentazoid! :smile:


Yes! You must use energy to find the speed …

how could you apply linear momentum when there's no linear motion and there's gravity?

But you will need Newton's second law to find when the normal force is zero.


V=Mgh = Mg(b+a)cos(theta) (h=0 at theta=90 degrees)

E=T+V

at top , ball is at rest
E(initial)=1/2*M*(0)^2+1/2*I*(0)+Mg(a+b)=Mg(a+b)

E(final)= 1/2*Mv^2+1/2*(2/5*Ma^2)(v/a)^2+ Mg(b+a)cos(theta)

E(i)=E(f) ==> Mg(b+a)=1/2*M*(7/5)v^2+Mg(b+a)cos(theta)

7/10*v^2==g(b+a)(1-cos(theta))

v^2=10/7* g(b+a)(1-cos(theta))

not sure how to find the angle between the line OG and the up ward vertical. How would I show that cos(theta)=10/17 when ball leaves sphere?

to get the normal force would I differentiate v

m*dv/dt=dP/dt=F=0

and I can now find theta?
 
  • #4
Hi pentazoid! :smile:

(have a theta: θ and a squared: ² :wink:)
pentazoid said:
V=Mgh = Mg(b+a)cos(theta) (h=0 at theta=90 degrees)

E=T+V

at top , ball is at rest
E(initial)=1/2*M*(0)^2+1/2*I*(0)+Mg(a+b)=Mg(a+b)

E(final)= 1/2*Mv^2+1/2*(2/5*Ma^2)(v/a)^2+ Mg(b+a)cos(theta)

E(i)=E(f) ==> Mg(b+a)=1/2*M*(7/5)v^2+Mg(b+a)cos(theta)

7/10*v^2==g(b+a)(1-cos(theta))

v^2=10/7* g(b+a)(1-cos(theta))

Very good! :biggrin:

(except most people use U for KE, since V looks too much like v :wink:)
not sure how to find the angle between the line OG and the up ward vertical. How would I show that cos(theta)=10/17 when ball leaves sphere?

to get the normal force would I differentiate v

Nooo … as I said, use Newton's second law …

what are the forces on the ball? …

they have to equal the centripetal acceleration …

so work out when the normal force becomes zero. :smile:
 

FAQ: Uniform ball rolling without slipping problem

What is the concept of "uniform ball rolling without slipping" problem?

The "uniform ball rolling without slipping" problem refers to a physics problem where a ball of a certain mass and radius rolls down a slope without slipping. This means that the ball maintains a constant velocity while also rotating at a constant angular velocity.

What are the key assumptions made in this problem?

The key assumptions made in this problem include: the ball is a perfect sphere, there is no friction between the ball and the surface, and there is no air resistance.

How does the solution to this problem differ from the solution to a regular "ball rolling down a slope" problem?

The solution to the "uniform ball rolling without slipping" problem takes into account the rotational motion of the ball, while the solution to a regular "ball rolling down a slope" problem only considers the translational motion of the ball. In the "uniform ball rolling without slipping" problem, the ball's velocity and angular velocity are constant, while in a regular "ball rolling down a slope" problem, the ball's velocity changes over time.

What is the role of the moment of inertia in this problem?

The moment of inertia, which is a measure of an object's resistance to rotational motion, plays a crucial role in the "uniform ball rolling without slipping" problem. It is used to calculate the ball's angular velocity and rotational kinetic energy in this problem.

Can this problem be applied to real-world situations?

Yes, the "uniform ball rolling without slipping" problem can be applied to real-world situations, such as the motion of a bowling ball or a rolling ball bearing. However, in reality, there will always be some amount of friction and air resistance present, so the problem will not be completely accurate.

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