On an amusement park ride, passengers are seated in a horizontal circle of radius 7.5 m. The seats begin from rest and are uniformly accelerated for 21 seconds to a maximum rotational speed of 1.4 rad/s.
-What is the instantaneous tangential speed of the passengers 15 s after the acceleration begins?
A)0.067 m/s
B)0.50 m/s
C)1.4 m/s
D)7.5 m/s
E)11 m/s

Question

Quizplus · Accepted Answer

To solve this problem, we need to use the equations of motion for circular motion.

First, we can use the equation for angular acceleration, which relates the final angular velocity, initial angular velocity, angular acceleration, and time:

ωf = ωi + αt

We know that ωi = 0 and ωf = 1.4 rad/s, and we are given that α = ωf/t = 1.4/21 = 0.067 rad/s^2. Plugging these values into the equation, we get:

1.4 = 0 + 0.067t

Solving for t, we get:

t = (1.4/0.067) = 20.9 s

So it takes about 20.9 seconds to reach the maximum speed of 1.4 rad/s.

Next, we can use the equation for tangential velocity, which relates the angular velocity and the radius of the circle:

v = ωr

At t = 15 s, we want to find the instantaneous tangential velocity, which means we need to plug in ω = αt (since the acceleration is constant during this time interval) and r = 7.5 m:

v = (0.067)(15)(7.5) = 7.54 m/s

Therefore, the instantaneous tangential speed of the passengers 15 s after the acceleration begins is approximately 7.5 m/s, which is closest to choice D.

On an Amusement Park Ride, Passengers Are Seated in a Horizontal