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f) A body is projected with velocity v up a plane inclined at an angle a to the ...

Aug 24, 2024

f) A body is projected with velocity v up a plane inclined at an angle a to the horizontal. When it returns through its starting point it is moving with half the speed with which is was projected. Determine the coefficient of friction u, in terms of the angle of the plane. Hint: The coefficient of friction is given by Ffriction = uN where N is the normal contact force and friction is the frictional force on the body.

Solution by Steps

step 1

Let the initial velocity be

v

and the angle of inclination be

a

. When the body returns to the starting point, its speed is

\frac{v}{2}

. The equations of motion can be used to analyze the forces acting on the body

step 2

The gravitational force acting down the incline is

mg \sin(a)

and the normal force is

N = mg \cos(a)

. The frictional force is given by

F_{\text{friction}} = uN = u(mg \cos(a))

step 3

Using energy conservation, the initial kinetic energy is

\frac{1}{2} mv^2

and the final kinetic energy when it returns is

\frac{1}{2} m\left(\frac{v}{2}\right)^2 = \frac{1}{8} mv^2

. The work done against friction and gravity must equal the loss in kinetic energy

step 4

The work done against gravity when moving up the incline is

mg h

where

h = d \sin(a)

and

d

is the distance traveled up the incline. The work done against friction is

F_{\text{friction}} \cdot d = u(mg \cos(a))d

. Setting up the equation:

mg h + u(mg \cos(a))d = \frac{1}{2} mv^2 - \frac{1}{8} mv^2

step 5

Simplifying the equation gives us

mg d \sin(a) + u(mg \cos(a))d = \frac{3}{8} mv^2

. Dividing through by

mgd

leads to

\sin(a) + u \cos(a) = \frac{3v^2}{8gd}

step 6

Rearranging for

u

gives

u = \frac{\frac{3v^2}{8gd} - \sin(a)}{\cos(a)}

. This expresses the coefficient of friction

u

in terms of the angle

a

Answer

u = \frac{\frac{3v^2}{8gd} - \sin(a)}{\cos(a)}

Key Concept

The relationship between kinetic energy, gravitational potential energy, and frictional forces in motion on an incline.

Explanation

The solution shows how to derive the coefficient of friction based on energy conservation principles and the forces acting on the body as it moves up and down the incline.