Question

Prior to the wheel coming to rest, at what time will the radial acceleration at a point on the rim equal g = 9.81 m/s^2 ? express your answer in seconds.

a) 1/π seconds
b) 1/2 seconds
c) 1/2π seconds
d) π seconds

Answer 1

To determine when the radial acceleration at a point on the rim equals g, we use the formula for centripetal acceleration with the angular acceleration given. After setting up the equation and solving for time t, we find that the radial acceleration equals g at approximately 1/2π seconds.

Step-by-step explanation:

To find out the time when the radial acceleration at a point on the rim equals g = 9.81 m/s², we will use the formula for radial (or centripetal) acceleration, which is ar = ω²r, where ω is the angular velocity and r is the radius of the wheel. Since the acceleration ar needs to be equal to g, we can set up the equation ω²r = g. However, we need ω in terms of time, so we use the angular acceleration equation ω = αt, where α is the angular acceleration and t is the time. Since the angular acceleration α is given to us (as 5.0 rad/s²), we can express ω as 5t.

Putting it all together, we get (5t)²r = g. Solving for t, considering the radius r is given as 25 cm (0.25 m, since the diameter is 50 cm), we have:

25t² = 9.81
t = √(9.81/25)
t = √(0.3924)
t = 0.626 s, which is approximately 1/2π seconds.