A sphere rolling without slipping down a hemisphere

In summary, the conversation discusses the impulse-momentum theorem, the angular impulse, energy considerations, conservation of energy, and the relationship between angular speed and the critical angle for a sphere sliding off a frictionless hemisphere. The conversation also addresses discrepancies in initial values and potential energy expressions, and suggests drawing a free body diagram to relate the angular speed to the critical angle. Additionally, there is a discussion about the flawed assumption of avoiding sliding with a horizontal impulse, and finding a simpler expression for the frictional impulse.
  • #1
lorenz0
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Homework Statement
A sphere of radius ##b##, mass ##m##, moment of inertia ##I_0## sits on the top of a hemisphere of radius ##a## with friction coefficient ##\mu##. An impulse ##J## is applied horizontally to the center of the sphere.
The friction ##\vec{F}_a## is such that the sphere can roll without slipping.
The sphere moves along the hemisphere until reaching angle ##\theta_D## (known), and at that point it loses contact with the hemisphere.
Find:

a) The relationship between the angular velocity of the sphere with respect to its center of mass ##\Omega## and the angular velocity of the center of mass of the sphere with respect to the center of the hemisphere ##\dot{\theta}##;

b) The angular velocity ##\Omega## and the kinetic energy just after the impulse is applied;

c) The angular velocity of the center of mass at ##\theta_D##;

d) Magnitude of the friction force for angles ##\theta>\theta_D##

e) The horizontal component of the velocity of the point of the sphere that touches the ground, at ##y=0##.
Relevant Equations
##\vec{J}=\Delta\vec{p}##, ##v=\Omega r##, ##E=K+U##, ##\vec{\tau}=\vec{r}\times\vec{F}=I\alpha##
a) From impulse-momentum theorem I have: ##J=mv## so ##v=\frac{J}{m}## and since the ball doesn't slip ##v=\Omega b## so ##\Omega=\frac{J}{mb}## and ##\dot{\theta}=\frac{v}{a+b}=\frac{\Omega b}{a+b}##.

b) I considered the angular impulse: ##-J(a+b)=I_0 \Omega_0 \Rightarrow \Omega_0=-\frac{J}{I_0}(a+b)##.
For the energy I have: ##\fbox{E}=mg(a+b)\sin(\theta)+\frac{1}{2}mv^2+\frac{1}{2}I_0 \Omega^2## and since the sphere is rolling without slipping ##v=\Omega b## hence ##E=mg(a+b)\sin(\theta)+\frac{1}{2}m(\Omega b)^2+\frac{1}{2}I_0 \Omega^2=mg(a+b)\sin(\theta)+\frac{1}{2}(I_0+mb^2)\Omega^2##.

c) Using conservation of energy I have: ##E_i=mg(a+b)+\frac{1}{2}(I_0+mb^2)\Omega_0^2, E_f=\frac{1}{2}\Omega_D^2 (I_0+mb^2)+mg(a+b)\sin(\theta_D)## so from ##E_i=E_f## I get ## \Omega_D=\sqrt{\Omega_0^2+\frac{2mg(a+b)(1-\sin(\theta_D))}{I_0+mb^2}} ##.

d) ##-bF_a=I_0 \alpha \Rightarrow |F_a|=|-\frac{I_0}{b}\frac{d\Omega}{dt}|##, where ##\Omega=\sqrt{\Omega_0^2+\frac{2mg(a+b)(1-\sin(\theta))}{I_0+mb^2}}\overset{(b)}{=}\sqrt{(-\frac{J}{I_0}(a+b))^2+\frac{2mg(a+b)(1-\sin(\theta))}{I_0+mb^2}}##.

e) ##v_x= - b\Omega_D## (negative since the rotation is clockwise)
-----

I am quite unsure about how I solved this problem so I would appreciate if someone would check it out and give me some feedback. Thanks.
 

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  • #2
Are you not disturbed by the different expressions for the initial value of ##\Omega## in parts (a) and (b). Which one do you think is correct and why?
You have a mixed bag of expressions for the potential energy that do not treat the position fo the CM of the sphere consistently. I recommend that, for the purpose of writing down the potential energy, you consider a point mass at distance ##(a+b)## from the center of the hemisphere.
You need to draw a free body diagram in order to relate the value of the angular speed to the critical angle ##\theta_D## when separation occurs.

These are my first impressions without having solved the problem. If you haven't solved the problem of a point mass sliding off a frictionless hemisphere, I suggest that you do that as a warm-up exercise.
 
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  • #3
kuruman said:
Are you not disturbed by the different expressions for the initial value of ##\Omega## in parts (a) and (b). Which one do you think is correct and why?
You have a mixed bag of expressions for the potential energy that do not treat the position fo the CM of the sphere consistently. I recommend that, for the purpose of writing down the potential energy, you consider a point mass at distance ##(a+b)## from the center of the hemisphere.
You need to draw a free body diagram in order to relate the value of the angular speed to the critical angle ##\theta_D## when separation occurs.

These are my first impressions without having solved the problem. If you haven't solved the problem of a point mass sliding off a frictionless hemisphere, I suggest that you do that as a warm-up exercise.
Thanks for the feedback. I think I have fixed the problem with the potential energy. I still don't see my mistake in part (a).
 
  • #4
lorenz0 said:
Thanks for the feedback. I think I have fixed the problem with the potential energy. I still don't see my mistake in part (a).
The question is a bit flawed. Given a horizontal impulse of arbitrarily short duration, there is no possibility that sliding was avoided. That would take a correspondingly arbitrarily large coefficient of friction. The sphere would have lost contact immediately.
However, if we assume such a coefficient initially, there must be a horizontal frictional impulse the other way from the hemisphere to achieve rolling. Maybe the author wants you to take that into account.
 
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  • #5
haruspex said:
The question is a bit flawed. Given a horizontal impulse, there is no possibility that sliding was avoided. That would take an infinite coefficient of friction. The sphere would have lost contact immediately.
However, if we assume such a coefficient initially, there must be a horizontal frictional impulse the other way from the hemisphere to achieve rolling. Maybe the author wants you to take that into account.
I see. Do you have any comments about the other parts of the problem? I am particularly concerned about part (d): I would like to find a less convoluted expression for ##F_a##.
 
  • #6
lorenz0 said:
I see. Do you have any comments about the other parts of the problem? I am particularly concerned about part (d): I would like to find a less convoluted expression for ##F_a##.
I have edited my post a little.

For d), I can't you just consider the forces on the sphere at that position, regardless of its angular velocity?
 
  • #7
haruspex said:
I have edited my post a little.

For d), I can't you just consider the forces on the sphere at that position, regardless of its angular velocity?
But if I consider the forces in the tangential direction won't the the angular velocity appear due to the presence of its derivative in Newton's Second Law? Namely, I get: ##mg \cos(\theta)-F_a=ma \alpha## (I use ##\cos(\theta)## since the angle is measure from the horizontal)
 
  • #8
lorenz0 said:
But if I consider the forces in the tangential direction won't the the angular velocity appear due to the presence of its derivative in Newton's Second Law? Namely, I get: ##mg \cos(\theta)-F_a=ma \alpha## (I use ##\cos(\theta)## since the angle is measure from the horizontal)
What if you consider a sphere placed at that position and released from rest? How would you calculate the frictional force then?
 
  • #9
haruspex said:
What if you consider a sphere placed at that position and released from rest? How would you calculate the frictional force then?
##F_a =-\mu N=-\mu mg\sin(\theta)##
 
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  • #10
Yes, for (d) draw an FBD of the sphere at that angle and figure out how much static friction is needed to provide the tangential acceleration. This part is the same as asking for the static friction when a ball is rolling down an inclined plane at fixed angle ##\theta_D##.
 
  • #11
lorenz0 said:
##F_a =-\mu N=-\mu mg\sin(\theta)##
Use the relationship between linear acceleration and angular acceleration, and the corresponding torque and force equations.
 
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  • #12
I believe that before the sphere has detached it must roll with slipping.

"The relationship between the angular velocity of the sphere with respect to its center of mass"

by definition, the angular velocity of a rigid body does not relate to any point.
 
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  • #13
wrobel said:
before the sphere has detached it must roll with slipping.
Good point.
 
  • #14
haruspex said:
Use the relationship between linear acceleration and angular acceleration, and the corresponding torque and force equations.
##mg\cos(\theta)-F_a=ma_t=mb\alpha,\ N-mg\sin(\theta)=ma_r,\ -bF_a=I_0 \alpha##; from the last equation I get ##\alpha=-\frac{bF_a}{I_0}## which, substituted into the first equation, gives ##F_a=\frac{mg\cos(\theta)}{1-\frac{mb^2}{I_0}}##.
 
  • #15
wrobel said:
I believe that before the sphere has detached it must roll with slipping.

"The relationship between the angular velocity of the sphere with respect to its center of mass"

by definition, the angular velocity of a rigid body does not relate to any point.
Interesting: why is that the sphere must roll with slipping before detaching from the hemisphere?
 
  • #16
lorenz0 said:
Interesting: why is that the sphere must roll with slipping before detaching from the hemisphere?
Because as it approaches the detachment point the normal force is tending to zero, so the available frictional force is also tending to zero. Since it is accelerating ever faster, the frictional force needed to maintain rolling is increasing.
 
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Related to A sphere rolling without slipping down a hemisphere

1. What is a sphere rolling without slipping down a hemisphere?

A sphere rolling without slipping down a hemisphere is a physical phenomenon in which a spherical object, such as a ball, is placed on top of a curved surface, such as a hemisphere, and begins to roll without sliding or slipping down the surface. This is due to the combination of the object's rotational and translational motion.

2. How does a sphere roll without slipping down a hemisphere?

A sphere is able to roll without slipping down a hemisphere due to the frictional force between the object and the surface. As the sphere rolls, the bottom point of contact experiences a static frictional force that prevents the sphere from slipping or sliding down the surface. This allows the sphere to maintain its rotational and translational motion without any loss of energy.

3. What factors affect the speed of a sphere rolling without slipping down a hemisphere?

The speed of a sphere rolling without slipping down a hemisphere is affected by several factors, including the mass and radius of the sphere, the angle of the hemisphere, and the coefficient of friction between the sphere and the surface. A larger mass and radius will result in a slower speed, while a steeper angle and higher coefficient of friction will result in a faster speed.

4. Can a sphere roll without slipping down a hemisphere indefinitely?

No, a sphere cannot roll without slipping down a hemisphere indefinitely. Eventually, the sphere will reach the bottom of the hemisphere and come to a stop due to the force of gravity. However, if the surface is frictionless, the sphere could potentially continue rolling without slipping indefinitely due to its inertia.

5. What are some real-life examples of a sphere rolling without slipping down a hemisphere?

One common example of a sphere rolling without slipping down a hemisphere is a ball rolling down a curved slide at a playground. Another example is a marble rolling down a curved track in a physics demonstration. This phenomenon can also be observed in sports, such as bowling or billiards, where the ball rolls without slipping on a curved surface.

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