A car is moving from rest .After 10 sec its wheel rotate 360 times in 1 minute. If the radius of the wheel is 50 cm, then find

(1) angular acceleration and
(2) angular velocity after 30 sec

To find the angular acceleration and angular velocity, we can use the equations of rotational motion.

Given:
Time, t = 10 seconds
Number of rotations, N = 360 rotations
Radius of the wheel, r = 50 cm

First, let's convert the radius to meters:
r = 50 cm = 0.5 meters

The number of rotations can be converted to radians:
360 rotations = 360 * 2π radians = 720π radians

Formula for angular acceleration (α):
α = (angular velocity final - angular velocity initial) / time

Formula for angular velocity (ω):
ω = angular displacement / time

Now, let's calculate the angular acceleration (α):

(1) angular acceleration (α):
We are given the initial angular velocity (ω₀) as the car is at rest. ω₀ = 0 rad/s

Using the formula α = (ω - ω₀) / t:

ω = 720π radians
t = 60 seconds (1 minute)

α = (720π - 0) / 60
α = 12π rad/s²

So, the angular acceleration is 12π rad/s².

Now, let's calculate the angular velocity after 30 seconds:

(2) angular velocity after 30 seconds (ω):
We'll use the formula ω = N * 2π / t, where N is the number of rotations and t is the time.

N = 720π radians (as calculated earlier)
t = 30 seconds

ω = 720π * 2π / 30
ω ≈ 151.48 rad/s

So, the angular velocity after 30 seconds is approximately 151.48 rad/s.

To find the angular acceleration, we need to use the formula:

Angular acceleration (α) = (Final angular velocity (ωf) - Initial angular velocity (ωi)) / Time (t)

Given that the wheel rotates 360 times in 1 minute, we can calculate the final angular velocity:

(1 minute = 60 seconds)
Angular velocity (ωf) = Number of rotations / Time
ωf = 360 rotations / 60 seconds
ωf = 6 rotations/sec

The initial angular velocity (ωi) is zero since the car is starting from rest.

Using the formula, we have:
α = (6 rotations/sec - 0 rotations/sec) / 10 sec
α = 0.6 rotations/sec²

Therefore, the angular acceleration is 0.6 rotations/sec².

To find the angular velocity after 30 seconds, we can use the formula:

Angular velocity (ω) = Initial angular velocity (ωi) + (Angular acceleration (α) × Time (t))

We know that the initial angular velocity is zero and the angular acceleration is 0.6 rotations/sec². The time is 30 seconds.

ω = 0 rotations/sec + (0.6 rotations/sec² × 30 sec)
ω = 18 rotations/sec

So, the angular velocity after 30 seconds is 18 rotations/sec.

wf=wi+alpha*time

360*2PI/60=0+alpha*60s
solve for alpha (rad/s^2)

wf=wi+alpha*time
wf=alpha*30

Notice you do NOT need the radius here, radius is needed to convert to linear velocity.