Measuring coefficients of friction. A coin is placed near one edge of a book lying on a table, and that edge of the book is lifted until the coin just slips down the incline. The angle of the incline Theta C, called the critical angle, is measured. (a) Draw a free body diagram for the coin when it is on the verge of slipping and identify all forces acting on it. Your free-body diagram should include a force of static friction acting up the incline. (b) Is the magnitude of the friction force equal to mu s n for the angles less than Theta C? Explain. What can you definitely say about the magnitude of the friction force for any angle Theta smaller or equal Theta C? (c) Show that the coefficient of static friction is given by mu s = tan Theta C. (d) Once the coin starts to slide down the incline, the angle can be adjusted to a new value Theta' C smaller or equal Theta C such that the coins moves down the incline with constant speed. How does observation enable you to obtain the coefficient of kinetic friction?

(a) To draw a free body diagram for the coin when it is on the verge of slipping, we need to consider all the forces acting on it.

- The weight of the coin acts downwards vertically. This force can be represented by its magnitude, which is the product of the mass of the coin (m) and the acceleration due to gravity (g).
Fg = m * g

- The normal force (N) acts perpendicular to the surface of the table and balances the weight of the coin. It can be represented by its magnitude, which is equal to the weight of the coin.
N = Fg = m * g

- The force of static friction (fs) acts along the incline in the opposite direction to the motion that would cause the coin to slip. This force prevents the coin from slipping down the incline until it reaches the critical angle.
fs

(b) The magnitude of the friction force is given by fs = μs * N, where μs is the coefficient of static friction and N is the normal force.

For angles less than θc (the critical angle), the coin is not slipping and the force of static friction is acting to prevent it from slipping. Therefore, the magnitude of the friction force (fs) must be equal to the product of the coefficient of static friction (μs) and the normal force (N).

(c) To show that the coefficient of static friction (μs) is given by μs = tan(θc), we can use trigonometry. In the free body diagram, we have the weight (mg) acting vertically downwards and the force of static friction (fs) acting up the incline. The angle of the incline is θc.

Using trigonometry, we can relate the magnitudes of these forces to the angle θc:
mg * sin(θc) = fs
mg * cos(θc) = N

Dividing the equation for fs by the equation for N:
(fs / N) = (mg * sin(θc)) / (mg * cos(θc))
μs = tan(θc)

Therefore, we have shown that the coefficient of static friction (μs) is given by μs = tan(θc).

(d) When the coin starts to slide down the incline, the static friction force is replaced by the kinetic friction force. For any angle θ smaller or equal to θc, the magnitude of the friction force will change to the coefficient of kinetic friction (μk) multiplied by the normal force (N).

To obtain the coefficient of kinetic friction (μk), we can adjust the angle to a new value θ'c smaller or equal to θc such that the coin moves down the incline with constant speed (zero acceleration). At this point, the magnitude of the friction force (fk) will be equal to μk * N.

By measuring the angle θ'c and knowing the coefficient of static friction (μs) from part (c), we can use the equation μk = tan(θ'c) to determine the coefficient of kinetic friction.