An automobile travelling at 80km per hours has tyres of radius 80 cm. 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 formula:
θ = ω₀ * t + (1/2) * α * t²
Where:
θ = number of turns made by the wheel (30 complete turns)
ω₀ = initial angular velocity of the wheel (0, as the car is brought to a stop)
t = time taken to bring the car to a stop (unknown)
α = angular acceleration of the wheel (unknown)
Plugging in the values we have:
30 = 0 * t + (1/2) * α * t²
30 = (1/2) * α * t²
We can simplify this equation by multiplying both sides by 2 to get:
60 = α * t²
Now, we need to find the value of t. We know that the car is brought to a stop and the wheels complete 30 turns. Since the car travels at a constant speed of 80 km/h, we can calculate the time taken to complete these 30 turns.
To find the time, we can use the formula:
time = distance / speed
The distance covered by the car while the brakes are applied is equal to the distance covered by one complete turn of the wheel multiplied by the number of turns (30).
distance = circumference of the wheel * number of turns
The circumference of the wheel can be found using the formula:
circumference = 2 * π * radius
Plugging in the values we have:
circumference = 2 * π * 80 cm
Now, since the question asks for the distance covered by the car while the brakes are applied, we need to convert the distance from centimeters to kilometers.
distance (in km) = distance (in cm) / 100,000
Now, we can calculate the time taken:
time = distance / 80 km/h
With the value of t, we can substitute it back into the equation:
60 = α * (time)²
Now we have an equation with α as the only unknown. Rearranging the equation:
α = 60 / (time)²
Substituting the value of time, we can calculate the magnitude of the angular acceleration of the wheels.
To find how far the car moves while the brakes are applied, you can use the formula for distance:
distance = speed * time
Plugging in the values:
distance = 80 km/h * time
Following the above steps will allow you to calculate the magnitude of the angular acceleration of the wheels and the distance the car moves while the brakes are applied.