An automobile traveling 80 mi/hr has wheels 31 inches in diameter. If the car is brought to a stop uniformly in 58 turns of the wheels, what is the magnitude of the angular acceleration of the wheels (in radians/second^2)?
C = pi * 2r = 3.14 * 31 = 97.4 In=8.1Ft.
d = 58rev * 6.28rad/rev = 364.24 Rad.
V=80mi/h*5280Ft/mi*6.28rad/8.1Ft*1h/3600s = 91 rad/s. = Vo.
V^2 = Vo^2 + 2a*d
a = (V^2-Vo^2)/2d
a = (0-(91)^2)/728.48 = -11.4 rad/s^2
To find the magnitude of the angular acceleration of the wheels, we first need to determine the initial and final angular velocities of the wheels.
Given:
- Speed of the automobile: 80 mi/hr
- Diameter of the wheels: 31 inches
- Time taken to stop uniformly: 58 turns of the wheels
First, let's convert the speed of the automobile from miles per hour to inches per second:
1 mile = 5,280 feet
1 foot = 12 inches
1 hour = 60 minutes
1 minute = 60 seconds
Therefore, the speed of the automobile can be calculated as follows:
Speed = (80 mi/hr) x (5280 ft/mi) x (12 in/ft) x (1 hr/60 min) x (1 min/60 sec) = 140.8 inches/sec
Next, let's calculate the distance covered by the wheels to bring the car to a stop.
Distance = Speed x Time = 140.8 inches/sec x 58 turns of the wheels
To find the circumference of the wheel, we use the formula:
Circumference = π x Diameter
The diameter of the wheel is given as 31 inches, so the circumference of the wheel is:
Circumference = π x 31 inches
Since the distance covered by the wheel equals the circumference times the number of turns, we can equate the two values:
(π x 31 inches) x (58 turns) = 140.8 inches/sec x 58 turns
Now, let's solve for π x 31 inches:
31π inches = 140.8 inches/sec
To calculate the value of π, divide both sides of the equation by 31 inches:
π = (140.8 inches/sec) ÷ (31 inches)
Now that we have the value of π, we can find the magnitude of the angular acceleration.
The formula to calculate angular acceleration is:
Angular acceleration = (Change in angular velocity) / (Change in time)
Since the car starts from rest and comes to a stop, the initial angular velocity is 0. The final angular velocity can be calculated using the distance and circumference of the wheel.
Final angular velocity = (Distance traveled) / (Circumference of the wheel)
Substituting the values we calculated earlier:
Final angular velocity = (140.8 inches/sec x 58 turns) / (π x 31 inches)
Finally, we can find the magnitude of the angular acceleration:
Angular acceleration = (Final angular velocity - Initial angular velocity) / (Time taken)
Since the initial angular velocity is 0, the equation simplifies to:
Angular acceleration = Final angular velocity / Time taken
Substituting the values:
Angular acceleration = ((140.8 inches/sec x 58 turns) / (π x 31 inches)) / 58 turns
Simplifying the equation gives us the magnitude of the angular acceleration of the wheels in radians/second².