What is the Acceleration of an Inclined Plane with a Mass on Top?

[ron_jay]

Homework Statement

An Inclined plane with an inclination of $$\theta$$ and a mass M has a body of mass m on top of it which is attached to a string; the string is light and passes over a pulley on top of the inclined plane and continues till it is firmly attached to a vertical wall.The friction coefficients between all the surfaces is zero or $$\mu$$ is 0.The block on the inclined plane moves down the slope as the inclined plane moves towards the right with an acceleration of 'a'.

Homework Equations

Calculate the acceleration of the inclined plane in terms of the two masses M and m.

The Attempt at a Solution

We see that the length of the string is constant. If we can differentiate the length with respect to time as it changes across the two sides of the pulley we might be able to the calculate the acceleration but I don't know how to apply that.

Component of gravity on mass 'm' :mgsin$$\theta$$ acting along the slope of the inclined plane

Tension 'T' on the string.

After that I really don't have a clue how to equate the forces to find the equations that will give the acceleration.Please Help.

 

[pardesi]

remeber the tension also acts on the pulley which itself is part of the system i.e. the inclined plane.
well there are two 'parts' of velocity of the bock one along the inclined palne and the other along the horizontal due to the horizontal motion of the inclined plane.
also the speed of the block along the inclined plane is same the incline's speed


but i would rather be using energy conservation here than getting into Newton' law

 

[ron_jay]

yes the tension does act on the pulley but aren't considering that at the moment. How would we equate it with energy conservation?

 

[pardesi]

the only l force which does work(net) here is that due to gravity.so apply work energy theorem u also can expree the kinetic energy in terms ofa single variable .differentiate both sides and u r done

 

[learningphysics]

Doesn't the tension in the string contribute external work?

I used sum of forces in the x and y direction for the small mass... with the sum of forces in the x direction for the big mass... Eliminated tension from the equations and solved for acceleration.

The main idea is to first relate the acceleration of the big block to the small one before using any force equations.

 

[pardesi]

no,the tension does no net work say the block moves $$dx$$ distance along the incline .then the incline moves $$dx$$ distance along the ground.
now the block has tension $$T$$ acting only along the incline .but the incline has 'two' tensions so net $$T-T \cos \theta$$ .so net work done is
on the block
$$T(-x) + T(x) \cos \theta$$

on the incline
$$T(1- \cos \theta)x$$

adding they cancel out

 

[learningphysics]

no,the tension does no net work say the block moves $$dx$$ distance along the incline .then the incline moves $$dx$$ distance along the ground.
now the block has tension $$T$$ acting only along the incline .but the incline has 'two' tensions so net $$T-T \cos \theta$$ .so net work done is
on the block
$$T(-x) + T(x) \cos \theta$$

on the incline
$$T(1- \cos \theta)x$$

adding they cancel out

Ah... you're absolutely right.

 

[ron_jay]

Is it so that the tension along the string is constantly changing or is it constant as the smaller mass moves down along the incline? Though i think not.

 

[pardesi]

well finally u realize that it is not changing after solving the problem but that's not a must

 

[learningphysics]

Is it so that the tension along the string is constantly changing or is it constant as the smaller mass moves down along the incline? Though i think not.

The tension turns out to be constant, but you don't have to assume anything about the tension to solve the problem. pardesi's method of using conservation of energy is the best way to solve the problem.

 

[ron_jay]

though I think using de Alembert's principle that the length of the string remains constant would be better.