A coin is placed on an old record turntable which rotates at 78 rpm. The coefficient of friction between the record and the coin is 0.17.

a) What is the angular speed of record in radians/s?
b) At what distance (m) from the center will the coin begin to slide off the record?
c) What is the period of the rotation (s) of the penny?

a) convert 78 rev/min to radians/sec. Call that number w.

b) R*w = 0.17 * M*g
Solve for radius, R.
c) Why would the penny rotate? It revolves about the turntable center before it starts slipping. It does not rotate while revolving.

To find the answers to these questions, we need to understand the relationships between angular speed, linear speed, and period in rotational motion.

a) Angular speed (ω) is defined as the rate of change of angular displacement with respect to time. In this case, the records rotate at 78 revolutions per minute (rpm). To convert this to radians per second (rad/s), we need to multiply by the conversion factor of (2π radians / 1 revolution) and (1 minute / 60 seconds).

ω = (78 rpm) * (2π radians / 1 revolution) * (1 minute / 60 seconds)

Simplifying the calculation:

ω = (78 * 2π / 60) rad/s

b) To find the distance from the center at which the coin will begin to slide off the record, we need to consider the balance of forces acting on the coin. The force of friction will equal the centripetal force required to keep the coin in circular motion.

The centripetal force (Fc) is given by the equation:

Fc = m * ω^2 * r

Where:
m = mass of the coin
ω = angular speed in rad/s, which we have calculated in part a)
r = distance from the center of the record

Since the coin is on the verge of sliding, the frictional force (Ff) will be equal to the maximum static frictional force (Ffmax) between the coin and the record. The maximum static frictional force is given by the equation:

Ffmax = μ * N

Where:
μ = coefficient of friction between the coin and the record (given as 0.17)
N = normal force between the coin and the record, which is equal to the weight of the coin (mg)

Setting Fc equal to Ffmax, we have:

m * ω^2 * r = μ * mg
ω^2 * r = μ * g
r = (μ * g) / ω^2

Substituting the given values, we can calculate the numerical value of r.

c) The period (T) of the rotation is the time required for one complete revolution. It is the inverse of the angular speed (ω).

T = 1 / ω

Substituting the calculated value of ω, we can find the period T.