A turntable rotates counterclockwise at 78 rpm. A speck of dust on the turntable is at 0.46 rad at t=0. What is the angle of the speck at t = 8.1 s? Your answer should be between 0 and 2π rad.

Answer: 3.79 rad

Step-by-step explanation:

Step 1 : Convert RPM to radians/second in order to work with the same units throughout the problem.

1 minute has 60 seconds.

Thus 78 RPM : 60 s = 1.3 Revolutions per second.

1 revolution = 2pi radians

1.3 RPS x 2pi = 8.16814 radians per second

Thus 78 RPM = 8.16814 rad/second

Step 2 : Calculate how far the dust speck travelled in 8.1 seconds.

We just calculated that the speck travels 8.16814 rad/s.

Therefore 8.16814 rad/s * 8.1 s = 66.162 rad/s

The dust speck travelled 66.162 radians in 8.1 s.

Step 3: Calculate the angle of the speck after 8.1 s

The dust peck was at 0.46 rad and travelled 66.162 rad thus 0.46+66.162 = 66.622 rad

Step 4: Convert to a value between 0 and 2pi radians.

Minus 10 revolutions in order to get to a value between 0 and 2pi radians.

66.622 - 10 (2pi) = 3.79 rad

Answer: The dust particle is at an angle of 3.79 rad