10. The drawing shows a baggage carousel at an airport. Your suitcase has not slid all the way down the slope and is going around at a constant speed on a circle (r = 12.4 m) as the carousel turns. The coefficient of static friction between the suitcase and the carousel is 0.75, and the angle θ in the drawing is 29.0°. How much time is required for your suitcase to go around once?

To find the time required for the suitcase to go around once, we need to consider the forces acting on the suitcase and the centripetal force that keeps it moving in a circular path.

The static friction acting between the suitcase and the carousel provides the required centripetal force to keep the suitcase moving in a circle. The centripetal force is given by the equation:

Fc = (m * v^2) / r

where Fc is the centripetal force, m is the mass of the suitcase, v is the velocity of the suitcase, and r is the radius of the circular path.

In this case, the static friction is equal to the maximum static friction, which is given by:

Fs = μ * N

where Fs is the static friction force, μ is the coefficient of static friction, and N is the normal force acting on the suitcase.

The normal force acting on the suitcase is equal to its weight, which is given by:

N = m * g

where g is the acceleration due to gravity.

Taking all these equations into account, we can calculate the centripetal force:

Fs = Fc

μ * N = (m * v^2) / r

μ * m * g = (m * v^2) / r

μ * g = (v^2) / r

v^2 = μ * g * r

v = √(μ * g * r)

Now, we can calculate the velocity of the suitcase using the given values. The coefficient of static friction, μ, is 0.75, the acceleration due to gravity, g, is approximately 9.8 m/s^2, and the radius of the circular path, r, is 12.4 m.

v = √(0.75 * 9.8 * 12.4)

v ≈ 8.628 m/s

The time required for the suitcase to go around once is equal to the circumference of the circular path divided by the velocity:

t = (2 * π * r) / v

t = (2 * π * 12.4) / 8.628

t ≈ 9.111 s

Therefore, it takes approximately 9.111 seconds for your suitcase to go around once on the carousel.