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?

420 rev/min = 420 * 2 pi /60 = 44 radians/second

120 rpm = 12.6 rad/s

alpha = (12.6 - 44 ) /4 = -7.86 radians/s^2 (part A)

angle = initial omega t + (1/2) alpha t^2
angle = 44 (4) - (1/2)(7.86)(16)
= 113 radians
113/2pi = 18 revolutions

18 * 2 pi r =18 *2 pi * .2 = 22.6 cm
but I think you mean meters, not centimeters

To solve this problem, we need to use the formulas for angular acceleration, angular displacement, and linear distance traveled. Let's break down each part of the problem and find the answers step by step.

A. What is the angular acceleration of the wheel?

The formula for angular acceleration is given by:

angular acceleration (α) = (final angular velocity - initial angular velocity) / time taken

Given:
initial angular velocity (ω1) = 420 rpm
final angular velocity (ω2) = 120 rpm
time taken (t) = 4 sec

First, we need to convert the angular velocities from rpm (revolutions per minute) to radians per second:
angular velocity in rad/s = angular velocity in rpm * 2π / 60

Converting the initial and final angular velocities:

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

Now, substitute the values into the formula for angular acceleration:

α = (ω2 - ω1) / t = (4π - 44π) / 4 = -40π / 4 = -10π rad/s²

Therefore, the angular acceleration of the wheel is -10π rad/s².

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

To find the angular displacement, we can use the formula:

angular displacement (θ) = (final angular velocity + initial angular velocity) / 2 * time taken

Substituting the values:

θ = (ω2 + ω1) / 2 * t = (4π + 44π) / 2 * 4 = 24π rad

Since there are 2π radians in one revolution, we can find the number of revolutions by dividing the angular displacement by 2π:

Number of revolutions = θ / 2π = 24π / 2π = 12 revolutions

Therefore, the angular displacement is 24π rad or 12 revolutions.

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

Here, we can use the relationship between angular displacement and linear distance:

linear distance (d) = angular displacement (θ) * radius

Substituting the values:

d = θ * r = 24π * 0.20 cm = 4.8π cm

Since π is approximately equal to 3.14, we can approximate:

d ≈ 4.8 * 3.14 cm ≈ 15.072 cm

Therefore, the car travels approximately 15.072 cm.