Find the acceleration : block slides on a wedge

[pcpssam]

sry for typo..it should be acceleration

Homework Statement

A block m slides down from a frictionless inclined surface (theta = 45) of a wedge, mass m, which is on a horizontal plane with coefficient of friction u. Prove that the wedge moves to the right with acceleration equal to g(1-3u)/(3-u) .

im stuck on this question for a whole week..but still can't get the answer
not sure if it is a question related to relative motion?
please help!

Homework Equations

The Attempt at a Solution

 

[Ibix]

As always, draw a force diagram.

If what you want is the acceleration of the wedge, you need to know the net force on the wedge. If you can't figure it out, post your working, and a photo of your diagram if possible.

 

[pcpssam]

cant get the relation of these three accelerations..

 

[Ibix]

You have three equations in four unknowns - you just need one more. You just need to enforce that the block stays on the wedge and doesn't go flying off - kinematic equations may help.

One minor point - your a_y has the same sign as g, but you have defined +y to be up the page. It doesn't affect your maths here, but it can do in general.

 

[Nickie]

To find the acceleration of the wedge, we can use Newton's Second Law, which states that the sum of all forces acting on an object is equal to its mass multiplied by its acceleration (ΣF = ma). We can break down the forces acting on the wedge into two components: the normal force (N) and the force of friction (Ff).

First, let's consider the forces acting on the block. Since it is sliding down a frictionless inclined surface, the only force acting on it is its weight (mg). We can break this force down into two components: mgcosθ acting perpendicular to the surface and mgsinθ acting parallel to the surface.

Next, let's consider the forces acting on the wedge. The normal force (N) acting on the wedge is equal to the weight of the block (mg) since the surface is frictionless. The force of friction (Ff) acting on the wedge is equal to uN, where u is the coefficient of friction between the wedge and the horizontal plane.

Using these forces, we can set up the following equations:

ΣFy = N - mgcosθ = 0 (since the wedge is not moving in the vertical direction)
ΣFx = Ff - mgsinθ = ma (since the wedge is moving in the horizontal direction)

Solving for N and Ff, we get:

N = mgcosθ
Ff = uN = umgcosθ

Substituting these values into the second equation, we get:

umgcosθ - mgsinθ = ma

Simplifying and rearranging, we get:

a = g(u - sinθ) / cosθ

Since θ = 45, we can simplify this further to:

a = g(u - 1/√2) / 1/√2

a = g√2(u - 1/2)

Finally, to find the acceleration of the wedge, we can use the fact that the force of friction (Ff) is equal to the mass of the wedge (m) multiplied by its acceleration (a):

Ff = ma

umgcosθ = ma

a = umgcosθ / m

a = ugcosθ

Substituting this into our previous equation, we get:

a = g√2(u - 1/2) = g(√2u - √2/