Its a Physics Question I am stuck with.?

Question

A disk s placed on a inclined plane with angle of inclination 'theta'. The disk is given an initial velocity of v(o) at an angle 'phi(o)'=90deg. where phi is the angle between the direction of motion of the disk and the x axis in the plane of the incline. The coefficient of friction is mu=tan(theta). The mass of the disk is m. Calculate the relation between the velocity v of the disk and the angle 'phi'.

Answer 1

This question is from Irodov i believe? 1.106 if i am not mistaken.

Here is how you do it. First of all understand that the magnitude of the frictional force will be mgsin(theta). Now this frictional force will be opposite to the direction of motion.

Write the equation of force balance for the x-direction and y-directon.(use mgsin(theta)cos(phi) and mgsin(theta)sin(phi) as the frictional force. write: frictional force+gravitational force=ma).

Let us call accelration in x as ax and in y as ay.

ax/ay=tan(phi) (might be otherway but you get the idea)

In the equation of force balance write ax as d(vx)/dt and d(vy)/dt...

write d(vx)/dt as d(vx)/d(phi)*d(phi)/dt.

Alse remember that vx/vy=tan(phi).

Devide the two equations and you will see that dt gets cancelled.

The differential equation becomes variable seperable.

A suggestion: Try not to ask irodov problems unless you have tried it in each and every possible manner. And don't buy and solution books of irodov as half of them are fakes with wrong sloutions.

Answer 2

Draw vector force diagram.

Vector force diagram is rhombus, the resulting net force is a bi-sector between velocity V and x-axis. Acceleration is tangent to the trajectory of velocity V in the space of vectors of velocities.

Because of this vector of velocity V follows parabolic trajectory in the space of velocities. Parabolic mirror reflects co-axial beam of light towards its focus V=0. The force of  gravity acts towards the directrix and the force of friction acts towards the focus V=0.

No differential equations are required here, nothing beyond ancient Greeks' knowledge (except relationship between velocity and acceleration and force)

Answer 3

If im am right then it is 1.1051