2. When traveling down the road at a constant speed of 55 mph, 26.4 m/s, the tangential velocity of the wheels is also 55 mph. If a car tire is 65.0 cm in diameter, then;
a. What is the period and frequency of the spinning car tire?
b. What is the centripetal acceleration of a rock stuck in the tire’s tread?
2. When traveling down the road at a constant speed of 55 mph, 26.4 m/s, the tangential velocity of the wheels is also 55 mph. If a car tire is 65.0 cm in diameter, then;
a. What is the period and frequency of the spinning car tire?
b. What is the centripetal acceleration of a rock stuck in the tire’s tread?
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To solve this problem, we need to understand the concepts of period, frequency, tangential velocity, and centripetal acceleration.
a. To find the period and frequency of the spinning car tire, we need to convert the tangential velocity from mph to m/s.
1. Convert the tangential velocity from 55 mph to m/s:
- 1 mile = 1609.34 meters
- 1 hour = 3600 seconds
- So, 55 mph = (55 * 1609.34) / 3600 = 24.58722 m/s
2. Calculate the period:
- The period (T) is the time it takes for one complete revolution of the tire.
- The circumference of the tire (C) can be calculated as:
- Circumference (C) = π * diameter
- Diameter = 65.0 cm = 0.65 m (converted from cm to m)
- C = π * 0.65 = 2.04159 m
- The period (T) can be calculated using the formula:
- T = C / tangential velocity
- T = 2.04159 / 24.58722 ≈ 0.083 s
3. Calculate the frequency:
- Frequency (f) is the number of complete revolutions per unit of time.
- The frequency (f) can be calculated using the formula:
- f = 1 / T
- f = 1 / 0.083 ≈ 12.048 Hz
b. To find the centripetal acceleration of a rock stuck in the tire's tread, we need to use the formula:
Centripetal acceleration = (tangential velocity)^2 / radius
- The radius can be calculated as half the diameter:
- Radius = 0.65 / 2 ≈ 0.325 m
- The centripetal acceleration can be calculated as:
- Centripetal acceleration = (24.58722)^2 / 0.325 ≈ 1487.04 m/s^2
So, the answers to the questions are:
a. The period of the spinning car tire is approximately 0.083 s, and the frequency is approximately 12.048 Hz.
b. The centripetal acceleration of a rock stuck in the tire's tread is approximately 1487.04 m/s^2.