A domesticated pig places a 100g rubber puck on a turntable. The turntable is set at a rate of 78rpm,and the pig knows the coefficient of static friction is 0.37 between the puck and turntable surface.
A) How far out from the center of the turntable can the puck be places before it spins off?
B) what difference would you expect from (A) if:
(i) the mass of the puck were doubled
(ii) the coefficient of static friction were doubled
To answer the question, we can use the concept of centripetal force and static friction. Let's break down the problem step by step:
A) How far out from the center of the turntable can the puck be placed before it spins off?
Step 1: Calculate the angular velocity
The angular velocity (ω) can be calculated by converting the given rpm into radians per second (rad/s):
ω = (2π × 78) / 60
Step 2: Calculate the tangential velocity
The tangential velocity (v) can be calculated by multiplying the angular velocity by the distance from the center of the turntable:
v = ω × r
Step 3: Calculate the maximum static friction force
The maximum static friction force (F_max) is equal to the product of the coefficient of static friction (μ) and the normal force (N), where N is equal to the weight of the puck:
F_max = μ × N
N = m × g, where m is the mass of the puck and g is the acceleration due to gravity.
Step 4: Equate the centripetal force to the maximum static friction force
For the puck to remain on the turntable, the centripetal force (F_c) must be less than or equal to the maximum static friction force:
F_c ≤ F_max
mv^2 / r ≤ μ × m × g
Step 5: Solve for the distance from the center
Rearranging the equation from Step 4, we can solve for the distance from the center (r):
r ≤ μ × g × v^2 / (g × μ)
r ≤ (v^2 / g)
Now, substitute the given values into the equation and calculate r.
B) What difference would you expect from (A) if:
(i) the mass of the puck were doubled?
In this case, the only difference would be in the calculation of the maximum static friction force (F_max). The weight of the puck (N) would change because it is directly proportional to the mass. So, you would need to recalculate F_max and re-evaluate the inequality in Step 4.
(ii) the coefficient of static friction were doubled?
If the coefficient of static friction (μ) is doubled, the maximum static friction force (F_max) would also double. As a result, the inequality in Step 4 would need to be modified by substituting the new value of μ and re-evaluating the calculation of r.