Question

A mustang going from 0 to 60 in 3.5 seconds has an acceleration of 7.7m/s. Radius of the tires is .35m, what is the angular displacement of the tires between the rest time and 60mph. Provide answer in radians and use the revolution of a tire is 2 x pie x r to convert to the linear distance that was traveled. Then also confirm this by using kinematics.

Answer 1

Answer on Question #48377, Physics, Mechanics | Kinematics | Dynamics

A mustang going from 0 to 60 in 3.5 seconds has an acceleration of $7.7\,\mathrm{m/s}$. Radius of the tires is $.35\,\mathrm{m}$, what is the angular displacement of the tires between the rest time and $60\,\mathrm{mph}$. Provide answer in radians and use the revolution of a tire is $2 \times$ pie $\times$ r to convert to the linear distance that was traveled. Then also confirm this by using kinematics.

Solution:

Given:

$$\begin{array}{l} v = 60\,\mathrm{mph} = 60 \times 0.44704 = 26.82\,\mathrm{m/s}, \\ t = 3.5\,\mathrm{s}, \\ a = 7.7\,\mathrm{m/s^2}, \\ r = 0.35\,\mathrm{m}, \\ \theta = ?, \end{array}$$

Angular displacement formula in terms of angular velocity is given by

$$\theta = \omega t$$

Where $\omega$ is angular velocity, $t$ is the time taken.

The average angular velocity is defined

$$\omega_{\text{average}} = \frac{\omega_0 + \omega_f}{2}$$

Linear speed = radius $\times$ angular speed

$$v = r \omega$$

Thus,

$$\omega_{\text{average}} = \frac{0 + \frac{v}{r}}{2} = \frac{v}{2r} = \frac{26.82}{2 \times 0.35} = 38.31\,\frac{\mathrm{rad}}{\mathrm{s}}$$

So,

$$\theta = 38.31 \times 3.5 = 134.1\,\mathrm{rad}$$

The linear displacement

$$S = \theta r = 134.1 \times 0.35 = 46.9 \approx 47\,\mathrm{m}$$

The angular displacement is the angle through which a body rotates in circular path. It is denoted by $\theta$ and expressed in radians

$$\theta = \frac{S}{r}$$

where $S$ is linear displacement.

Kinematics equation

$$S = \frac{v^2 - v_0^2}{2a}$$

where $a$ is acceleration, $S$ is distance, $v_0$ is initial velocity and $v$ is final velocity.

Thus,

$$S = \frac{26.82^2}{2 \times 7.7} = 46.71 \approx 47\,\mathrm{m}$$

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