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

[MatinSAR]

Concentrate on the ball rather than the ramp. What are the individual forces acting on the ball?

You calculated the net work done on the ball in your first post. This net work on the ball must equal the sum of the works done by the individual forces acting on the ball.

Any kind of air resistance and friction is ignored. Do you mean another force?!

What are the individual forces acting on the ball?

mg and N.

 

[TSny]

mg and N.

OK.
What is the numerical value of the work done by mg on the ball?
What is the numerical value of the total work done on the ball?
What can you deduce from this concerning the work done by N on the ball?

 

[MatinSAR]

What is the numerical value of the work done by mg on the ball?

9 J

What is the numerical value of the total work done on the ball?

8 J

What can you deduce from this concerning the work done by N on the ball?

-1 J

But the work done by N must be 0, isn't it perpendicular to the direction of movement?

 

[jbriggs444]

But the work done by N must be 0, isn't it perpendicular to the direction of movement?

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

 

[TSny]

9 J

8 J

-1 J

Good

But the work done by N must be 0, isn't it perpendicular to the direction of movement?

No. At each instant, N is perpendicular to the surface of the ramp. But the direction of motion of the ball at some instant is not parallel to the surface of the ramp. Play around with some sketches showing the initial and final positions of the ball and ramp for a small time interval. To get the basic idea, consider a ramp in the shape of an inclined plane.

 

[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.