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 ?
circumfance=PI*r^2=PI*(.65/2)^2
period=cir/dpeed=above/26.4 sec
freq= 1/period
ang freq=2PI*freq
centripetal acc= w^2*radius
To find the period and frequency of the spinning car tire, we need to use the formula:
Period (T) = 1 / Frequency (f)
a. To find the period:
Convert the tangential velocity from mph to m/s:
55 mph = 55 * 0.44704 m/s ≈ 24.587 m/s
The circumference of the tire can be calculated using the formula:
Circumference (C) = π * diameter
C = π * 65.0 cm ≈ 204.203 cm ≈ 2.04203 m
The time taken for one complete rotation (period) is given by:
T = 2π * r / v
Where r is the radius of the tire and v is the tangential velocity.
Convert the radius from cm to meters:
65.0 cm = 0.65 m
T = 2π * 0.65 m / 24.587 m/s ≈ 0.167 s
b. To find the centripetal acceleration of a rock stuck in the tire's tread, we can use the formula:
Centripetal acceleration (a) = (tangential velocity)^2 / radius
a = (24.587 m/s)^2 / 0.65 m ≈ 934.417 m/s^2
Therefore:
a. The period of the spinning car tire is approximately 0.167 s and the frequency is the reciprocal of the period: f = 1 / T ≈ 1 / 0.167 ≈ 5.988 Hz.
b. The centripetal acceleration of a rock stuck in the tire's tread is approximately 934.417 m/s^2.
To calculate the period and frequency of the spinning car tire, we can use the formula:
Period (T) = 1 / Frequency (f)
The frequency is the number of complete revolutions the tire makes in one second, and the period is the time it takes for one complete revolution.
a. First, let's find the time taken for one complete revolution of the tire. The circumference of the tire can be calculated using the formula:
Circumference = π * Diameter
Given that the tire's diameter is 65.0 cm, the circumference will be:
C = π * 65.0 cm
C ≈ 204.20 cm
Since the car is traveling at a constant speed of 55 mph, or 26.4 m/s, we can calculate the time taken for one revolution using the formula:
Time = Circumference / Tangential Velocity
Given that the tangential velocity of the wheels is 26.4 m/s, the time taken for one revolution will be:
Time = 204.20 cm / 26.4 m/s
To ensure that we have the same units, let's convert the circumference to meters:
C = 2.042 m
Time = 2.042 m / 26.4 m/s
Time ≈ 0.0777 s
Now, we can calculate the frequency:
Frequency (f) = 1 / Time
f ≈ 1 / 0.0777 s
f ≈ 12.89 Hz
Therefore, the period of the spinning car tire is approximately 0.0777 seconds, and the frequency is approximately 12.89 Hz.
b. To calculate the centripetal acceleration of a rock stuck in the tire's tread, we can use the formula:
Centripetal Acceleration (a) = (Tangential Velocity)^2 / Radius
Given that the tangential velocity of the wheels is 26.4 m/s, and the radius of the tire can be calculated by dividing the diameter by 2:
Radius = 65.0 cm / 2
Radius = 32.5 cm = 0.325 m
Now, we can calculate the centripetal acceleration:
a = (26.4 m/s)^2 / 0.325 m
a ≈ (696.96 m^2/s^2) / 0.325 m
a ≈ 2141.8 m/s^2
Therefore, the centripetal acceleration of a rock stuck in the tire's tread is approximately 2141.8 m/s^2.