Work and Energy for a mass sliding down a curved ramp (why does the ramp move?)

[MatinSAR]

Not if the surface exerting the normal force $N$ is moving.

Can you explain how is it negative ?

 

[jbriggs444]

Can you explain how is it negative ?

The component of the surface's motion in the direction of the normal force is negative. It is moving away from the force it exerts. So the force pair does positive work on the surface and negative work on the ball.

 

[TSny]

Can you explain how is it negative ?

For a small time interval, you will need to imagine the displacement of the ball. Does this displacement have a component parallel or antiparallel to the normal force acting on the ball?

 

[MatinSAR]

Does this displacement have a component parallel or antiparallel to the normal force acting on the ball?

N is neither perpendicular nor parallel to displacement.
As the object descends and the surface moves to the left, the object tilts slightly downward and does not move parallel to the surface.

Am I wrong?!

 

[TSny]

N is neither perpendicular nor parallel to displacement.
As the object descends and the surface moves to the left, the object tilts slightly downward.

But does the displacement of the ball have a component parallel (or antiparallel) with N?

The picture below shows a small interval of time with initial time in black and final time in brown for the case of an inclined plane for the ramp.

If the ball were to move only parallel to the plane, then it would end up at the position shown by the open circle. But the ball must end up on the brown ramp. The overall displacement of the ball is from the intial black position of the ball to the final brown position of the ball.

 

[MatinSAR]

But does the displacement of the ball have a component parallel (or antiparallel) with N?

antiparallel ??

 

[TSny]

Yes. Good. So, do you see how N does negative work on the ball?

 

[MatinSAR]

Yes. Good. So, do you see how N does negative work on the ball?

N and one of the displacement components are in the opposite direction, so the work is negative.
Is it true ??

 

[TSny]

N and one of the displacement components are in the opposite direction, so the work is negative.
Is it true ??

Yes.

 

[MatinSAR]

Yes.

Your answer was one of the best I have ever received, even better than my professors at university. Thank you for your valuable time.

 

[TSny]

Thank you for the kind comment.

 

[MatinSAR]

Right. If there is potential energy to consider, then the conserved quantity is mechanical energy which is the sum of kinetic and potential energy of the two-mass system. In other words, the sum of the kinetic energies of the masses plus the potential energy when the mass is at some point on the ramp does not change as the mass descends and the ramp moves to the left.

Thanks a lot.

The component of the surface's motion in the direction of the normal force is negative. It is moving away from the force it exerts. So the force pair does positive work on the surface and negative work on the ball.

Thank you for your help.