Write an algebraic expression for the coefficient of kinetic friction for a block of mass m sliding down an incline of angle theta with respect to the horizontal with acceleration a. (type the formula out with the word (theta) for the angle. Make sure you are careful with your parentheses. You may use the following variables: m, a, g, (theta) and the sin, cos, and tan function

n = {gsin (theta) - a}/gcos (theta)

OR n = tan (theta) {at constant velocity}

Explanation:

For a block of mass m sliding down an incline at angle theta, the forces acting on the object along the plane are the moving force (down the plane) and the frictional force which is acting opposite the moving force (i. e acting up the plane).

The moving force Fm = Wsin theta where W is the weight of the block and theta is the angle of inclination.

The frictional force Ff is expressed as the product of the coefficient of kinetic friction (n) and the normal reaction force (R) acting on the body i. e Ff = nR

Since the body is accelerating down the plane, we will apply the Newton's second law to solve for the coefficient of the kinetic friction.

According to the law;

Force = mass * acceleration

F = ma ... (1)

Summation of the force acting on the body along the plane will be;

Fm - Ff

= Wsin (theta) - nR

Since R = Wcos (theta), the sum of the forces becomes;

Wsin (theta) - n (Wcos (theta))

Since W = mg, the equation becomes;

mgsin (theta) - nmgcos (theta) ... (2)

Substituting equation 2 into 1, we will have;

mgsin (theta) - nmgcos (theta) = ma

gsin (theta) - ngcos (theta) = a

ngcos (theta) = gsin (theta) - a

Dividing both sides by gcos (theta) to get n which is the coefficient of kinetic friction will give;

n = {gsin (theta) - a}/gcos (theta)

If the body moves with a constant velocity, acceleration 'a' will be zero and the coefficient of kinetic friction n will become;

n = {gsin (theta) - 0}/gcos (theta)

n = gsin (theta) / gcos (theta)

n = sin (theta) / cos (theta)

n = tan (theta)

uk = tan thetha

Explanation:

Since the block slides at constant velocity, a=0 hence

Mg sin tetha = Fr = ma since a=0

Mg sin thetha = Fr

But Fr = is the horizontal force acting on the body and Mg cos thetha is the normal reaction

But Fr = uk N

Where uk is the coefficient of kinetic friction between the block and the plane

uk = Fr/N = Mg sin thetha / Mg cos thetha

uk = tan thetha