Mr John is driving his car along the road. The radius of his car tyres are 0.28 m. suddenly 60 m in front of his car his angry wife stand still trying to stop him. He decelerates at 7.0 m per second to avoid hitting his dear wife. The wheel angular velocity before the deceleration was 100 rads per second. Calculate the angular acceleration of the tyre of john's car assuming they do not slip on the pavement.
Calculate: (a) the angular acceleration of the tyres assuming they do not slip on the pavement. (b). Calculate the number of revolutions the tyres make before coming to a rest. (c). Calculate the amount of time it takes a car to stop completely. (d). Calculate the distance travelled during the time calculated in part c above.
To calculate the angular acceleration of the car's tires, we need to use the formula:
Angular acceleration (α) = (final angular velocity (ωf) - initial angular velocity (ωi)) / time (t)
Given information:
- Initial angular velocity (ωi) = 100 rads/s
- Final angular velocity (ωf) = 0 rads/s (as the tire comes to a complete stop)
- Time (t) = ?
To find the time, we first need to calculate the linear distance covered by the car during the deceleration:
Distance = 60 m (given)
Since the car is coming to a stop, the linear displacement is equal to the circumference of the tire:
Circumference (C) = 2πr
= 2 * 3.14159 * 0.28 m
= 1.76079 m
Now, we can calculate the time taken (t):
Distance = circumference * number of rotations
= C * (ωi * t)
Rearranging the equation, we have:
t = Distance / (C * ωi)
Plugging in the values:
t = 60 m / (1.76079 m * 100 rads/s)
= 0.3409 s
Now that we have the time, we can calculate the angular acceleration using the formula mentioned earlier:
Angular acceleration (α) = (ωf - ωi) / t
= (0 - 100 rads/s) / 0.3409 s
= -293.274 rads/s^2
Therefore, the angular acceleration of John's car's tires is approximately -293.274 rads/s^2.