An automobile travelling at 80km/hrs has tyres of radius 80cm. On applying brakes the car is brought to a stop in 30 complete turns of the tyres. What is the magnitude of the angular acceleration of the wheels? How far does the car move while the brakes are applied?
To find the magnitude of the angular acceleration of the wheels, we can use the following formula:
Angular acceleration (α) = (final angular velocity (ωf) - initial angular velocity (ωi)) / time (t)
Given that the car is brought to a stop in 30 complete turns of the tyres, we can find the time taken (t) to come to a stop.
Given that the car is traveling at 80 km/hr, which is equal to 80,000 m/3600 s, we can calculate the initial angular velocity (ωi) using the formula:
ωi = linear velocity (v) / radius (r)
Since the linear velocity (v) is measured in m/s, we need to convert the initial velocity to m/s and the radius to meters as well.
Given that the radius of the tires is 80 cm, which is equal to 0.8 meters, we can calculate the initial angular velocity (ωi):
ωi = 80,000 m/3600 s / 0.8 m = 27.78 rad/s
Now, to find the final angular velocity (ωf), we can use the formula:
ωf = 0 rad/s (because the car comes to a stop)
Now, with the time taken (t) and the initial and final angular velocities (ωi and ωf), we can calculate the angular acceleration (α):
α = (0 rad/s - 27.78 rad/s) / t
To find the distance the car moves while the brakes are applied, we can use the formula:
Distance = (initial velocity (v) * time (t)) - (0.5 * angular acceleration (α) * time (t)^2)
Let's plug in the calculated values and solve for the answers.