Find Force of Friction on Wedge: Shm Physics Help

[chaoseverlasting]

Homework Statement

A wedge of mass M is placed on a rough surface with a mass m attached to a spring placed rigidly on the wedge. The surface b/w the wedge and the mass m is smooth and the surface between the wegde and the ground is sufficiently rough to keep the wedge from moving. The angle of inclination of the wedge w.r.t ground is 60 degrees. The small block is displaced by a small distance x. Find the force of friction between the ground and the wedge.

Homework Equations

$$\frac{d^2 x}{dt^2)=-kx$$
$$x=asin(\omega t)$$

The Attempt at a Solution

Force acting on the mass m :

$$F=-kx$$
Force along the horizontal direction:
$$F=-kxcos(60)$$
$$F=-\frac{1}{2}kx$$
$$m\frac{d^2x}{dt^2}=-\frac{1}{2}kx$$
$$\frac{d^2x}{dt^2}=-\frac{1 k}{2 m}x$$
$$a=-\frac{k}{2m}asin(\omega t)$$
$$\frac{k}{m}=\omega ^2$$
$$f_r=Ma$$
$$f_r=\frac{1 Ma\omega ^2sin(\omega t)}{2}$$

but the answer given is $$f_r=\frac{ma\omega ^2sin(\omega t)}{2}$$
Is the book wrong or have I made a mistake?

 

[cristo]

Are you using a to mean two different things here?

 

[chaoseverlasting]

Yeah, sorry. In the end it should be A, for amplitude.