At a playground, a 20-kg child sits on a spinning merry-go-round, as shown from above in (Figure 1) . The merry-go-round completes one revolution every 6.2 s, and the child sits at a radius of r=1.8m.

Question

At a playground, a 20-kg child sits on a spinning merry-go-round, as shown from above in (Figure 1) . The merry-go-round completes one revolution every 6.2 s, and the child sits at a radius of r=1.8m.

a) What is the force of static friction acting on the child?
b) What is the minimum coefficient of static friction between the child and the merry-go-round to keep the child from slipping?

Answer 1

force = mass * centripetal acc

= m v^2/r

c = 2 pi r = 2 pi * 1.8
v = (c/6.2) meters/second

so
F = 20 * (c/6.2)^2/1.8

when F = mu * m g, we slip
mu = Ac/g= (c/6.2)^2/(1.8*9.81)

Answer 2

To find the force of static friction acting on the child, we need to consider the centripetal force acting on the child.

a) The centripetal force is given by the formula:

F_c = m * v^2 / r

Where,
F_c is the centripetal force,
m is the mass of the child (20 kg),
v is the linear velocity of the child, and
r is the radius of the circular path (1.8 m).

The linear velocity can be calculated by dividing the distance traveled in one revolution (the circumference of the circle) by the time period of one revolution:

v = 2πr / T
= 2π(1.8 m) / 6.2 s

Substituting the values into the centripetal force formula, we can calculate:

F_c = (20 kg) * [2π(1.8 m) / 6.2 s]^2 / (1.8 m)

b) The minimum coefficient of static friction required to keep the child from slipping can be found by considering the maximum force of static friction that can be exerted. This force is equal to the force of static friction acting on the child at its maximum.

The maximum force of static friction is given by the equation:

F_friction_max = μ_s * F_n

Where,
F_friction_max is the maximum force of static friction,
μ_s is the coefficient of static friction, and
F_n is the normal force (equal to the weight of the child).

The normal force can be calculated as:

F_n = m * g

where,
m is the mass of the child (20 kg), and
g is the acceleration due to gravity (approximately 9.8 m/s^2).

Substituting the values into the equation for the maximum force of static friction, we get:

F_friction_max = μ_s * (20 kg) * 9.8 m/s^2

So, to find the minimum coefficient of static friction, we can rearrange the equation:

μ_s = F_friction_max / (m * g)

Now, we can substitute the calculated value of F_friction_max and the given values of mass and acceleration due to gravity into the equation to solve for the coefficient of static friction, μ_s.