The tires of a car make 63 revolutions as the car reduces its speed uniformly from 94.0 km/h to 58.0 km/h. The tires have a diameter of 0.88 m. What was the angular acceleration of each tire? And If the car continues to decelerate at this rate, how much more time is required for it to stop? I didn't understand how to do this problem when someone on here already tried to explain, so could someone please explain step-by-step!
To find the angular acceleration of each tire, we can use the formula:
angular acceleration (α) = (final angular velocity - initial angular velocity) / time
1. Convert the speeds from km/h to m/s:
94.0 km/h = 94.0 * (1000 m / 1 km) / (3600 s / 1 h) ≈ 26.1 m/s
58.0 km/h = 58.0 * (1000 m / 1 km) / (3600 s / 1 h) ≈ 16.1 m/s
2. Calculate the initial and final angular velocities:
The number of revolutions of the tires is not needed for this part, only their diameters. The circumference of each tire can be calculated using the formula:
circumference = π * diameter
Initial circumference = π * 0.88 m ≈ 2.77 m
Final circumference = π * 0.88 m ≈ 2.77 m
The initial angular velocity (ω_i) can be calculated by dividing the initial speed by the initial circumference:
ω_i = 26.1 m/s / 2.77 m ≈ 9.4 rad/s
The final angular velocity (ω_f) can be calculated by dividing the final speed by the final circumference:
ω_f = 16.1 m/s / 2.77 m ≈ 5.8 rad/s
3. Calculate the time taken to decelerate:
Since the problem describes that the car reduces its speed uniformly, we can assume that the angular acceleration (α) is constant throughout the process. We'll use the formula:
α = (ω_f - ω_i) / t
We can rearrange this formula to solve for time (t):
t = (ω_f - ω_i) / α
4. Substitute the given values into the equation:
The problem does not directly provide the value for α, but we can calculate it using the formula:
α = (ω_f - ω_i) / t
= (5.8 rad/s - 9.4 rad/s) / t
Let's solve for α first:
α = (5.8 rad/s - 9.4 rad/s) / t
Since the problem does not provide the value for α, we cannot directly compute the time.
To find the angular acceleration of each tire, we need to use the formula:
angular acceleration (α) = (change in angular velocity) / (change in time)
Step 1: Convert the speed from km/h to m/s
94.0 km/h = 94.0 * (1000/3600) m/s = 26.11 m/s
58.0 km/h = 58.0 * (1000/3600) m/s = 16.11 m/s
Step 2: Calculate the change in angular velocity
Change in angular velocity = Final angular velocity - Initial angular velocity
The angular velocity can be calculated using the formula:
angular velocity = speed / (radius of the tire)
radius of the tire = diameter / 2
radius of the tire = 0.88 m / 2 = 0.44 m
Initial angular velocity = 26.11 m/s / 0.44 m = 59.34 rad/s
Final angular velocity = 16.11 m/s / 0.44 m = 36.61 rad/s
Change in angular velocity = 36.61 rad/s - 59.34 rad/s = -22.73 rad/s (negative sign indicates deceleration)
Step 3: Calculate the change in time
To find the change in time, we can use the formula:
time = (number of revolutions * circumference of the tire) / speed
Number of revolutions = 63
Circumference of the tire = 2 * π * radius of the tire = 2 * π * 0.44 m
Initial time = (63 * (2 * π * 0.44 m)) / 26.11 m/s = 6.13 s
Final time = (63 * (2 * π * 0.44 m)) / 16.11 m/s = 16.62 s
Change in time = 16.62 s - 6.13 s = 10.49 s
Step 4: Calculate the angular acceleration
angular acceleration (α) = (change in angular velocity) / (change in time)
angular acceleration (α) = (-22.73 rad/s) / (10.49 s)
α ≈ -2.17 rad/s²
The angular acceleration of each tire is approximately -2.17 rad/s².
If the car continues to decelerate at this rate, we can find the time required for it to stop using the formula:
time = |final angular velocity / angular acceleration|
Final angular velocity = 0 rad/s (since the car is stopping)
Angular acceleration = -2.17 rad/s²
time = |0 rad/s / -2.17 rad/s²|
time ≈ 0 s
Therefore, no additional time is required for the car to stop since the initial angular acceleration will cause the car to stop instantaneously.