Question

easy problem-solving help a car is traveling with a speed of 20.0 m/s along a straight horizontal road. the wheels have a radius of 0.300 m. if the car speeds up with a linear acceleration of 1.50 m/s² for 8.00 s, find the angular displacement of each wheel during this period.

Answer 1

The angular displacement of each wheel is found using the kinematic equation for angular displacement. Calculating the initial angular velocity and substituting values yield the result for the given linear acceleration and time.

To find the angular displacement
`(\(\theta\))` of each wheel during the acceleration period, we can use the kinematic equation relating linear acceleration
`(\(a\))`, initial velocity
`(\(v_0\))`, time
`(\(t\))`, and angular displacement:

`\[ \theta = \omega_0 t + (1)/(2) a t^2 \]`

where:

• 
  `\(\omega_0\)` is the initial angular velocity,

• 
  `\(a\)` is the linear acceleration,

• 
  `\(t\)` is the time.

First, calculate the initial angular velocity
`(\(\omega_0\))` using the linear velocity and wheel radius:

`\[ v_0 = \omega_0 r \]`

`\[ \omega_0 = (v_0)/(r) \]`

Given:

- Linear velocity,
`\(v_0 = 20.0 \ \text{m/s}\)`

- Wheel radius,
`\(r = 0.300 \ \text{m}\)`

`\[ \omega_0 = \frac{20.0 \ \text{m/s}}{0.300 \ \text{m}} \]`

Now, substitute the known values into the angular displacement formula:

`\[ \theta = \left((v_0)/(r)\right)t + (1)/(2) a t^2 \]`

Given:

• Linear acceleration,
  `\(a = 1.50 \ \text{m/s}^2\)`

• Time,
  `\(t = 8.00 \ \text{s}\)`

`\[ \theta = \left(\frac{20.0 \ \text{m/s}}{0.300 \ \text{m}}\right) \cdot 8.00 \ \text{s} + (1)/(2) \cdot 1.50 \ \text{m/s}^2 \cdot (8.00 \ \text{s})^2 \]`

Calculate the angular displacement
`\(\theta\)`.