Question

Two wooden blocks are moving on a smooth horizontal surface such that the mass m remains stationary with respect to block of mass M as shown in the figure. The magnitude of force P is:

• A. (M + m)g tan
• B. g tan
• C. mg cos
• D. (M + m) g cosec

Answer 1

The correct option is A

(M + m)g tan

If the force is P and the block is in rest with respect to the wedge that means both the wedge and the block have the same acceleration.

Okay this means

Acceleration a = $\frac{P}{TotalMass}=\frac{P}{(M+m)}$----------------(i)

Now that we know that both the wedge and block move with acceleration a, if I go in wedge's frame and try making the Free Body Diagram of block apart from mg and normal I'll have to add another component called pseudo force so as newton's laws work.

You know by now that pseudo force is applied to object in non-inertial frame and in opposite direction of acceleration of the non-inertial frame. Here wedge is accelerating rightwards. So pseudo force on block is leftwards. It is equal to m $\times$ acceleration of the frame so ma

Now if I am in wedge frame. In layman terms if I am sitting on the wedge and observing the block then with respect to me the block isn't moving right.

So with respect to me its in equilibrium

$Fe{t}_{net}$ = 0

$\sum F_y=0$

N cos B - mg = 0

N cos B = mg

N = $\frac{mg}{cosB}$ -----------------(i)

$\sum F_x=0$

N sin B - ma = 0

N sin B = ma

mg tan B = ma

From equation (i)

g tan B = $\frac{p}{(M+m)}$

p = (M + m) g tan B