A problem about momentum and energy,

[garylau]

1. Homework Statement
A small mass m slides without friction on a surface making a quarter-‐circle with radius R, as shown. Then it lands on the top surface of a cart, mass M, that slides without friction on a horizontal surface. (In practice, this cart could be a slider on an air-‐track.) Between the top of the cart and the mass m, the coefficient of kinetic friction is µk. The mass m slides a distance d along the top of the cart, but doesn't fall off. (i) When there is no longer any relative motion between m and M, how fast is the cart.Show all working and assumptions and state carefully and explicitly any relevant laws or principles. (Hint: you will find it helpful to break the problem up into separate stages and to draw diagram for each.

(ii) Determine how far the mass m slides along the surface of the cart before stopping on it. State explicitly any relevant laws or principles and any relevant approximations.

Homework Equations

mv1+Mv2=(m+M)v

F=ma 1/2mv^2=mgh

The Attempt at a Solution

i have done the first part successfully and i try to do the second part as following

but the answer is wrong

the correct answer of the (ii) part is : d= {R(M/(m+M)µk}

when i try to use D1-D2 to get the total distance between the cart and the mass
it looks messy

Any other way to do this question?

thank

 

[TSny]

Your work looks correct. Before subtracting D₁ - D₂, you might try simplifying D₁ to one fraction.

 

[garylau]

Your work looks correct. Before subtracting D₁ - D₂, you might try simplifying D₁ to one fraction.

thank

but this way looks to complicated

Do you have other method to do this question

thank
for example: do it in the centre of mass frame or using the conservation of energy??

 

[TSny]

It's not that complicated. You've already done most of the work for this method of solving the problem.

Yes, you can approach the problem using energy concepts. It will take less calculation. Was there any loss of mechanical energy from the time block m was released and the time when m and M move together? If so, how much mechanical energy was lost?

 

[garylau]

It's not that complicated. You've already done most of the work for this method of solving the problem.

Yes, you can approach the problem using energy concepts. It will take less calculation. Was there any loss of mechanical energy from the time block m was released and the time when m and M move together? If so, how much mechanical energy was lost?

If i use the centre of mass frame to do this question

Will i got it easily?

 

[TSny]

For me, it seems to take about the same amount of effort to solve it in the CM frame. But I could be overlooking something.

 

[garylau]

For me, it seems to take about the same amount of effort to solve it in the CM frame. But I could be overlooking something.

using the centre of mass frame

we can let the total final KE =0 because both object moves with the centre of mass

and just find the initial KE ?

 

[TSny]

using the centre of mass frame

we can let the total final KE =0 because both object moves with the centre of mass

and just find the initial KE ?

Yes. Just find the initial KE in the CM frame as m lands on M.