Final answer:
To prevent the cat from sliding off the merry-go-round, the necessary coefficient of static friction is calculated using the angular velocity, the radius of merry-go-round, and the acceleration due to gravity. The centripetal force, which depends on angular velocity and radius, must be countered by the force of static friction, derived from the cat's mass and gravity.
Step-by-step explanation:
Calculating the Necessary Coefficient of Static Friction
To determine the least coefficient of static friction that would allow the cat to stay in place on the merry-go-round, we need to use the formula for centripetal force and static friction. The centripetal force, which is required to keep the cat moving in a circle, is provided by the force of static friction between the cat and the surface of the merry-go-round.
First, we calculate the angular velocity (ω) of the merry-go-round: ω = 2π / T, where T is the period of rotation (6.00 s). Then, we can express the centripetal acceleration (a_c) as a_c = ω² × r, where r is the radius of the merry-go-round (5.40 m). The force of static friction (f_s) must be equal to or greater than the mass of the cat times the centripetal acceleration (f_s ≥ m × a_c) to prevent the cat from sliding.
Since centripetal acceleration is directed towards the center of the circle and the force of static friction is directed opposite to the potential motion of sliding (tangentially to the circle's edge), the coefficient of static friction (μ_s) can be found by the equation f_s = m × g × μ_s. After rearranging the formula, we solve for μ_s to be the least coefficient of static friction needed: μ_s = a_c / g, where g is the acceleration due to gravity.
By plugging in the values for a_c and g, we determine the minimum coefficient of static friction required to keep the cat stationary relative to the merry-go-round as it turns.