A wheel on a moving car slows uniformly from 420 rpm to 120 rpm in 4 sec.

A. What is the angular acceleration of the wheel?

B. What is the angular displacement? How many revolutions?

C. How far does the car travel if the radius of the wheel is 0.20cm?

angular acceration=(120-420)/4 in rpm/sec

now to change that to radians/sec^2
multipy by 2PI/60 to get rad/sec^2

To solve this problem, we need to use the formulas for angular acceleration, angular displacement, and linear displacement.

Let's start with part A, which asks for the angular acceleration of the wheel. We know that angular acceleration (α) is defined as the change in angular velocity (ω) divided by the change in time (t):

α = (ω2 - ω1) / t

In this case, the initial angular velocity (ω1) is given as 420 rpm, and the final angular velocity (ω2) is given as 120 rpm. We converted these angular velocities from rpm to rad/s, using the conversion factor of (2π rad)/(60 s):

ω1 = 420 rpm * (2π rad/60 s) = 44π rad/s
ω2 = 120 rpm * (2π rad/60 s) = 4π rad/s

Plugging these values into the formula, we get:

α = (4π - 44π) rad/s / 4 s = -40π rad/s²

Therefore, the angular acceleration of the wheel is -40π rad/s² (negative value indicating deceleration).

Moving on to part B, which asks for the angular displacement. Angular displacement (θ) is defined as the change in angular position. In this case, we know the initial and final angular velocities and the time, so we can use the average angular velocity (ω_avg) formula to find the angular displacement:

ω_avg = (ω1 + ω2) / 2

Plugging in the given values, we get:

ω_avg = (44π rad/s + 4π rad/s) / 2 = 24π rad/s

Now, we can use the formula for angular displacement using average angular velocity:

θ = ω_avg * t

Plugging in the values, we get:

θ = 24π rad/s * 4 s = 96π rad

To find the angular displacement in terms of revolutions, we can convert from radians to revolutions, using the conversion factor of 1 revolution = 2π radians:

θ_in_revolutions = 96π rad * (1 revolution / 2π rad) = 48 revolutions

Therefore, the angular displacement of the wheel is 96π radians or 48 revolutions.

Finally, for part C, to find the linear displacement of the car, we can use the formula relating linear displacement (s) to angular displacement (θ) and wheel radius (r):

s = θ * r

Given that the radius of the wheel is 0.20 cm, we need to convert it to meters:

r = 0.20 cm * (1 m / 100 cm) = 0.002 m

Plugging in the values, we get:

s = 96π rad * 0.002 m = 0.192π m

Therefore, the car travels approximately 0.192π meters.