# Mechanics Problem-Please help!

1. Nov 30, 2004

### Trigger

Please help me out with the following problem. I have only just begun a mechanics course and am finding it extremely difficult!

(a)A cart with initial mass M and a load of sand 1/2M loses its sand at the rate k kg/sec. the cart is puled horizontally by a force F. Find the differential equation for the rate of change of the cart's velocity v in terms of k, M and F while there is sand in the cart?

(b)Suppose F=0 and the cart's initial velocity is Vo. What is the sand's final velocity when all the sand has left the cart?

(c)A box is filled with sand and is sliding down a smooth plane. The sand is leaking out at a constant rate. If the plane's inclination is A, what is the equation of motion of the box plus all the sand?

2. Nov 30, 2004

### HallsofIvy

Staff Emeritus
Force= rate of change of momentum= d(mv)/dt

You probably know F= ma which is a special case when m is a constant.

In general F= d(mv)/dt= m(dv/dt)+ v(dm/dt).

(a) You are told that m starts out as (3/2)M but decreases at constant rate k:
m= (3/2)M- k and dm/dt= -k.

That means F= ((3/2)M-k)(dv/dt)- kv. That is the differential equation you want.

(b) Now take F= 0 and solve the differential equation for v(t). You will need to determine what t is when all the sand is gone.

(c) Almost the same as (b). The force directly downward is mg but you are on a plane with angle A. Can you find the "component of force" along the plane?

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