Sunny have been appointed to an amusement ride safety committee for the Mall of America's Nickelodeon Universe, which is reviewing the safety of a ride that consists of seats mounted on each end of a rotating steel beam. For most of the ride, the beam rotates about its center in a horizontal circle at a constant speed. One committee member insists that a person moving in a circle at constant speed is not accelerating, so there is no need to be concerned about the ride's safety. Another thinks that the person has a constant acceleration when moving at a constant speed. Yet a third argues that the person's acceleration depends on the rate of change of their velocity, not their speed. Since each component of the person's velocity changes with time, their acceleration must change with time. He decide to settle the issue by making a model of the ride and measuring the magnitude of the acceleration of different positions on the model when it spins at a constant speed.

Equipment: apparatus that spins a horizontal platform, stopwatch, meterstick.

1. Make a drawing of the path of an object in circular motion at constant speed. On that path, use a dot to represent the object's position at time t₁. Label this point as O, and draw a vector at O to represent the magnitude and direction of the object's velocity at time t₁. Draw another dot to represent the object's position at a later time t₂, shortly after t₁, and label this point P. Draw a vector at P to show the magnitude and direction of the object's velocity at time t₂.
2. Redraw the velocity vectors with the tail of one vector (point P) at the tail of the other vector (point O). Keep the same size and direction as in the previous drawing. To find the acceleration of the object that is interested in the change in velocity (v). The change vis the increment that must be added to the velocity at time t₁ so that the resultant velocity has the new direction after the elapsed time t=t1t2. Add the change in velocity v to the drawing of the velocity vectors; it should be a straight line connecting the heads of the vectors.
3. On the drawing from question 1, label the distance r from the center of the circle to points O and P. In the limit that the time interval is very small, the arc length distance traveled by the object can be approximated as a straight line. Use this approximation to label the distance traveled by the object along the circle from point O to P in terms of the object's velocity and the elapsed time.
4. The triangle drawn in question 2 (with v and v) is similar to the triangle drawn in question 3 (with r and the straight line distance traveled by the object) because they have the same apex angle. Use the relationship of similar triangles to give an equation that connects the sides and the bases of the two triangles.
5. Solve the equation for v/t to get an expression for the acceleration in terms of the object's uniform velocity and the distance r.
6. From the equation, is the acceleration of an object in circular motion ever zero? Does the magnitude of the acceleration change with time?

Prediction

Does an object moving in a circle accelerate? If so, does the magnitude of the acceleration change with time? Explain the reasoning. Use the acceleration equation derived in the Warm-up to support the claim