Sphere striking an incline (not asking for solutions....)

In summary, the sphere is released at a height H above a fixed inclined plane, and the coefficient of restitution at impact is e>0. The sphere leaves the surface just after impact, and the coefficient of friction between the sphere and the plane is \mu.
  • #1
PhMichael
134
0
The sphere is released at a height [itex]H[/itex] above a fixed inclined plane, as shown in the attached figure.
The coefficient of restitution at impact is [itex]e>0[/itex] (that is the sphere leaves the surface just after impact), the coefficient of friction between the sphere and the plane is [itex]\mu[/itex].

I need a clarification to what happens during impact.

Is it even possible for the sphere to roll without slipping during the very short time interval of impact with the inclined plane?

I would assume that since the velocity component in the tangent direction [itex]\boldsymbol{t}[/itex] of the impact point on the sphere (B) just before impact is nonzero, specifically it equals [itex]\sqrt{2gH} \sin(\alpha)[/itex], the sphere would necessarily slip on the inclined plane with the impulsive friction force pointing in the negative [itex]\boldsymbol{t}[/itex] direction and having the magnitude [itex]|\hat{f}| = \mu |\hat{N}|[/itex] (a superposed hat denotes an impulse).

Using the balance equations of linear and angular impulse-momentum together with the definition of the coefficient of restitution, it is possible to obtain the velocity of the center of mass of the sphere and its angular velocity. However, if a no-slip condition during impact is imposed, then there is another constraint on the velocity of the impact point (B) on the sphere, which, in my opinion, is not a possible scenario. Am I right?
 

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  • #2
Not sure why you think it must slip. As you say, there is impulsive friction acting up the plane. Just do the impulse balance, as you would a force balance in statics.
 
  • #3
haruspex said:
Not sure why you think it must slip. As you say, there is impulsive friction acting up the plane. Just do the impulse balance, as you would a force balance in statics.

I obviously did that.
If I assume that the ball does not slip during impact then this means that its normal velocity vanishes, which is not true because it leaves the surface.
Am I getting something wrong here?
 
  • #4
PhMichael said:
I obviously did that.
If I assume that the ball does not slip during impact then this means that its normal velocity vanishes, which is not true because it leaves the surface.
Am I getting something wrong here?
There is impulsive torque. It "instantly" gains rotation.
 

1. What is the concept behind a sphere striking an incline?

The concept behind a sphere striking an incline is the interaction between forces and motion. When a sphere rolls down an incline, it experiences a combination of gravitational force and frictional force, which affects its motion and speed.

2. How does the angle of incline affect the motion of the sphere?

The angle of incline plays a crucial role in determining the speed and acceleration of the sphere. A steeper incline will result in a faster and more accelerated motion, while a shallower incline will result in a slower and less accelerated motion.

3. What factors influence the distance a sphere travels when striking an incline?

The distance a sphere travels when striking an incline is affected by various factors such as the initial speed of the sphere, the angle of incline, and the coefficient of friction between the sphere and the incline surface.

4. How does the mass of the sphere affect its motion on an incline?

The mass of the sphere has a minimal effect on its motion when striking an incline. As long as the sphere is rolling without slipping, its mass does not affect its acceleration or speed. However, a heavier sphere may experience a greater force of gravity, which can affect its speed and distance traveled.

5. Can we accurately predict the motion of a sphere striking an incline?

With the knowledge of the initial conditions and the laws of physics, we can accurately predict the motion of a sphere striking an incline. However, in real-world scenarios, other factors such as air resistance and imperfections in the incline surface may affect the accuracy of our predictions.

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