# A problem about momentum and energy,

• garylau
In summary, the problem involves a small mass sliding on a quarter-circle and landing on a cart with a coefficient of kinetic friction between them. The first part of the problem involves finding the speed of the cart when there is no longer any relative motion between the small mass and the cart. The second part involves determining how far the small mass slides on the surface of the cart before stopping. This can be solved using the method of subtracting D1 from D2 or by using energy concepts in the centre of mass frame.
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.

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

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

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

garylau
TSny said:
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??

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?

TSny said:
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?

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.

TSny said:
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 ?

garylau said:
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.

## 1. What is momentum and energy?

Momentum is a measure of an object's motion, while energy is the ability to do work. In physics, momentum is defined as the product of an object's mass and its velocity, while energy is the capacity to do work or cause change.

## 2. How are momentum and energy related?

Momentum and energy are both conserved quantities, meaning they cannot be created or destroyed. In a closed system, the total amount of momentum and energy will remain constant. Additionally, momentum and energy can be interchanged through various processes, such as collisions or transformations.

## 3. What is the equation for calculating momentum?

The equation for calculating momentum is p = mv, where p is momentum (in kg*m/s), m is mass (in kg), and v is velocity (in m/s). This equation shows that an object with a greater mass or velocity will have a greater momentum.

## 4. How does kinetic energy relate to momentum?

Kinetic energy is the energy an object possesses due to its motion. It is directly proportional to the mass and velocity of an object, meaning that an object with greater momentum will also have greater kinetic energy. In fact, the equation for kinetic energy is KE = 1/2 * mv^2, which includes both mass and velocity.

## 5. Can momentum and energy be lost?

No, momentum and energy are conserved quantities and cannot be lost. In a closed system, the total amount of momentum and energy will remain constant, and they can only be transferred between objects or transformed into different forms.

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