Question

7) A wheel rotates through an angle of 320° as it slows down from 78.0 rpm to 22.8 rpm. What is the magnitude of the average angular acceleration of the wheel? Ans: 5.48 rad/s2

Answer 1

The magnitude of the average angular acceleration for the wheel is 5.48 rad/s².

Step-by-step explanation:

The magnitude of average angular acceleration can be calculated using the formula:

Angular acceleration (α) = (change in angular velocity) / (change in time)

In this case, the initial angular velocity = 78.0 rpm, and the final angular velocity = 22.8 rpm. To convert these values to rad/s, multiply by (2π/60).

So, initial angular velocity = 78.0 rpm * (2π/60) rad/s = 8.1848 rad/s
Final angular velocity = 22.8 rpm * (2π/60) rad/s = 3.7808 rad/s

Now, we can substitute these values into the formula:

α = (3.7808 rad/s - 8.1848 rad/s) / t

Given that the angle of rotation is 320°, we can convert it to radians by multiplying by (π/180) as follows:

Angle of rotation (θ) = 320° * (π/180) = 5.585 rad

Substituting the values into the formula, we get:

α = 5.585 rad / t

To solve for t, we can use the formula:

t = (θ) / (α)

Substituting the values, we get:

t = 5.585 rad / (5.585 rad / t)

Simplifying, we find t = 1 second.

Therefore, the magnitude of the average angular acceleration is 5.48 rad/s² (rounded to two decimal places).

Answer 2

The question involves finding the average angular acceleration of a wheel as it slows down, which requires converting angular speeds from rpm to rad/s, determining the time of deceleration and solving kinematic equations.

Step-by-step explanation:

The question asks for the magnitude of the average angular acceleration of a wheel that slows down from 78.0 revolutions per minute (rpm) to 22.8 rpm as it rotates through an angle of 320°.

Steps to Calculate Average Angular Acceleration:

1. Convert the initial and final angular speeds from rpm to radians per second (rad/s):
   ω_i = 78 rpm × (2π rad/1 rev) × (1 min/60 s) = 8.168 rad/s
   ω_f = 22.8 rpm × (2π rad/1 rev) × (1 min/60 s) = 2.387 rad/s
2. Calculate the change in angular velocity (Δω):
   Δω = ω_f - ω_i = -5.781 rad/s
3. Determine the time taken for the change in velocity, assuming uniform angular acceleration:
   θ = 320° = 320° × (π rad/180°) = 5.585 rad
   Equation: θ = ω_i t + (1/2) α t^2
4. Solve for the time (t) and then find the average angular acceleration (α).

The calculations involve knowledge of angular kinematics and unit conversions typically covered in a high school physics course.