the car has wheels of radius 0.3 m and is travelling at a speed of 36 m/s in a straight line and the wheel now describes 40 revolution while the car is brought to rest;
calculate the wheels angular acceleration.
φ=ω₀t-εt²/2
ω =ω₀-εt
ω=0
ω₀=v/r
φ=2πN
ε =ω₀²/2φ =v²/2r²•2πN=
= v²/4πr²N=36²/4•π•0.3²•40=28.6 rad/s²
To calculate the wheel's angular acceleration, you need to determine the initial and final angular velocities of the wheel, as well as the time it takes for the car to come to a complete stop.
First, let's calculate the initial and final angular velocities:
Initial angular velocity (ω₁) can be calculated using the formula:
ω₁ = v₁ / r
where v₁ is the initial linear velocity and r is the radius of the wheel.
Given:
v₁ = 36 m/s
r = 0.3 m
Substituting the values, we get:
ω₁ = 36 / 0.3 = 120 rad/s
To find the final angular velocity (ω₂), we can use the formula for angular displacement:
θ = ω₁t + (1/2)αt²
where θ is the total angular displacement, α is the angular acceleration, and t is the time taken.
The car comes to rest, so its final angular displacement (θ) is given by the number of revolutions (N) completed by the wheel:
θ = 2πN
Given:
N = 40 revolutions
θ = 2π(40) = 80π rad
Since the car starts from rest and comes to a complete stop, the final angular velocity (ω₂) is zero.
Using the formula for angular displacement, we have:
0 = ω₁t + (1/2)αt²
Substituting the initial angular velocity (ω₁) and solving for α, we get:
(1/2)αt² = -ω₁t
αt² = -2ω₁t
α = -2ω₁ / t
Now, we need to find the time taken (t) for the car to come to a complete stop. We can use the formula for linear deceleration, assuming the car decelerates uniformly to zero velocity:
v₂ = v₁ + at
0 = 36 + a * t
a = -36 / t
We know that the linear velocity (v₂) is equal to the tangential velocity at the wheel's rim, which is given by:
v₂ = ω₂ * r
Since v₂ = 0 (car is at rest), ω₂ * r = 0.
So we have:
0 = ω₂ * 0.3
Since ω₂ = 0, the time taken (t) can be calculated as:
0 = -36 / t
t = -36 / 0
t = 0
Now, substituting t = 0 into α = -2ω₁ / t, we get:
α = -2ω₁ / 0
Therefore, the wheels' angular acceleration (α) is undefined. This means that the wheels do not experience any angular acceleration or deceleration during the process.