# What is the angular velocity of the nail in a car tire at 55 mi/hr?

• Miike012
In summary, to find the angular velocity of the nail in rad/hour, we need to first convert the car's velocity from 55 miles per hour to inches per hour. Then, using the formula v=r(θ/t), we can calculate the angular speed by dividing the car's velocity by the radius of the wheel. Finally, to find the angular velocity, we multiply the angular speed by the radius.
Miike012

## Homework Statement

Car runs at 55 mi/hr. and the nail is lodged in tire 13 in from center of wheel, then what is the angular velocity of the nail in rad/hour?

w= v/r

## The Attempt at a Solution

First I will convert 55 miles into inches.
55 mile * 5280 FT/mile * 12 Inches/FT = 3484800 inches.

Velocity = 3484800 INCH/HR
R = 6.5 INCH

Thus:3484800 / 6.5 = 6969600 inch/hr
(Is this correct so far)...
then how do I convert that into rad/hr...?
Is there a formula that I am not aware of?

The formula for angular velocity is v=r(θ/t) where r is the radius, t is time, and theta is the angle in radians, though you can use degrees if I'm not mistaken

But I am not given theta. Not that I am aware of anyways... I am only given 55 mi/hr and the radius.

actually you are given theta, it is the angle of the wheel, or 2pi radians

O yeah.. so do i multiply 2(pi) times 55 mi/hr / R (in miles)?

not quite. You know that the angle is 2pi radians you need to figure out how long it take to sweep across that distance, in other words how long does it take for the wheel to make one rotation which you can find out since you know how fast the car itself is travelling. θ/t gives you your angular speed. you then multiply by the radius to find the angular velocity.

## 1. What is angular velocity of a wheel?

Angular velocity of a wheel is a measure of how fast the wheel is rotating around its axis. It is expressed in radians per second (rad/s) or revolutions per minute (rpm).

## 2. How is angular velocity of a wheel calculated?

The formula for calculating angular velocity is ω = Δθ / Δt, where ω is the angular velocity, Δθ is the change in angular displacement, and Δt is the change in time.

## 3. What factors affect the angular velocity of a wheel?

The angular velocity of a wheel is affected by its radius, the applied torque, and the moment of inertia of the wheel. Other factors such as friction and air resistance may also have an impact.

## 4. Can the angular velocity of a wheel change over time?

Yes, the angular velocity of a wheel can change over time if there is a change in the applied torque or if there is a change in the moment of inertia of the wheel. It can also change due to external factors such as friction or air resistance.

## 5. Why is the angular velocity of a wheel important?

The angular velocity of a wheel is important because it is a key factor in determining the speed and motion of the wheel. It is also a fundamental concept in understanding rotational motion and is used in many real-world applications such as vehicle design, robotics, and machinery.

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