a car has wheel of 30cm radius. it starts from rest and accelerates uniformly to a speed of 15 m/s in a time of 8s. find the angular acceleration of its wheels and the number of rotations one makes in this time.
v₀ = 0
v = ω R
ω=v/R =15/0.3=50 rad/s
ω =εt=>
ε = ω/t= 50/8 =6.25 rad/s²
2πN = εt²/2
N= εt²/4π=6.25•8²/4π= 31.8 rev
Well, if the car's wheel has a radius of 30cm, that means it has a diameter of 60cm or 0.6m.
We know that the car starts from rest and accelerates uniformly to a speed of 15 m/s in 8 seconds. In order to find the angular acceleration of the wheels, we can use the formula:
ω = Δv / r
Where ω is the angular acceleration, Δv is the change in linear velocity, and r is the radius of the wheel.
The change in linear velocity is given by:
Δv = final velocity - initial velocity
Δv = 15 m/s - 0 m/s
Δv = 15 m/s
Now, let's calculate the angular acceleration:
ω = Δv / r
= 15 m/s / 0.3 m
= 50 rad/s^2
So, the angular acceleration of the car's wheels is 50 rad/s^2.
To find the number of rotations the wheel makes in this time, we can use the formula:
θ = ωt
Where θ is the angle (in radians), ω is the angular acceleration, and t is the time.
θ = 50 rad/s^2 * 8 s
= 400 rad
Since one rotation is equal to 2π radians, the number of rotations the wheel makes in this time would be:
Number of rotations = θ / (2π)
= 400 rad / (2π)
≈ 63.66 rotations (approximately)
So, the wheel makes approximately 63.66 rotations in this time.
Now that was a wheely good question!
To find the angular acceleration of the car's wheels, we can use the formula:
Angular acceleration (α) = (Final angular velocity - Initial angular velocity) / Time
We know that the car starts from rest, so its initial angular velocity is 0.
Angular acceleration (α) = (15 m/s) / 8s
Calculating this, we get:
α = 1.875 rad/s²
To find the number of rotations the wheel makes in 8 seconds, we need to find the total angle covered by the wheel.
Total angle covered = Angular velocity × Time
We know that angular velocity is related to linear velocity by the formula:
Angular velocity (ω) = Linear velocity (v) / Radius (r)
Using this formula, we can find the angular velocity:
ω = v / r
= 15 m/s / 0.3 m
= 50 rad/s
Now, we can calculate the total angle covered:
Total angle covered = (Angular velocity) × (Time)
= (50 rad/s) × (8s)
= 400 radians
Since one complete rotation is equal to 2π radians, we can find the number of rotations:
Number of rotations = Total angle covered / (2π)
= 400 radians / (2π)
≈ 63.66 rotations
Therefore, the angular acceleration of the car's wheels is approximately 1.875 rad/s², and the number of rotations the wheel makes in 8 seconds is approximately 63.66 rotations.
To find the angular acceleration of the car's wheels, we will use the formula:
angular acceleration (α) = (final angular velocity (ω) - initial angular velocity (ω₀)) / time (t)
First, we need to find the initial angular velocity and the final angular velocity.
The initial angular velocity (ω₀) is 0 since the car starts from rest.
The final angular velocity (ω) can be calculated using the formula:
ω = v / r
where v is the linear velocity and r is the radius of the wheel.
Given that v = 15 m/s and r = 30 cm = 0.3 m, we can calculate ω:
ω = 15 m/s / 0.3 m
ω = 50 rad/s
Now, we can find the angular acceleration (α) using the formula:
α = (ω - ω₀) / t
Since ω₀ = 0, the formula simplifies to:
α = ω / t
Plugging in the values:
α = 50 rad/s / 8 s
α = 6.25 rad/s²
Therefore, the angular acceleration of the car's wheels is 6.25 rad/s².
To find the number of rotations the wheel makes in this time, we can use the formula:
number of rotations = final angular displacement / (2π)
The final angular displacement can be calculated using the formula:
θ = ω₀t + (1/2)αt²
Given that ω₀ = 0 and α = 6.25 rad/s², we can calculate θ:
θ = (1/2)(6.25 rad/s²)(8 s)²
θ = 200 rad
Plugging this value into the formula for the number of rotations:
number of rotations = 200 rad / (2π)
number of rotations ≈ 31.83 rotations
Therefore, the wheel makes approximately 31.83 rotations in a time of 8 seconds.