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Linear Speed

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Linear path refers to the plain path like a straight road. You could often be walking, riding, running in a straight line road. The speed you are carrying is what linear speed is! So lets see the concept of linear speed and how to determine linear speed and angular speed in circular motion and linear motion.

Linear Speed Definition

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The Linear speed is that speed what the body has when its in a straight line path. It is the rate of change of linear displacement for a time t. Suppose a body displaces from a point along a circumference at a radius r and the ray from the center of circle to the point traverses angle $\theta$ radians in time t.
Linear Speed
Then its Linear speed is given by
Linear Speed v = $\frac{S}{t}$
Here s is the linear displacement and t is the time taken.

Linear Speed Formula

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There are two formulas for linear speed:
The linear speed in terms of linear displacement is given by
v = $\frac{\Delta\ s}{\Delta\ t}$
Here S represents linear displacement and t is the time taken.

The angular speed in terms of linear speed is given by
v = $\omega$ r
Where $\omega$ is the angular speed and r is the radius of circular path.

Rotational Speed to Linear Speed

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Linear speed is the rate at which the linear distance is traveled. The rotational speed (Angular speed) is the rate at which the rotations are performed. Linear and rotational speed are related a lot with each other. In fact while undergoing angular displacement $\theta$ each time the body will be in linear path.
Referring figure above you could measure angle as
$\theta$ = $\frac{Arc\ length}{time}$ = $\frac{s}{t}$Differentiate w.r.to time t to get
$\frac{d \theta}{dt}$ = $\frac{1}{r}$ $\frac{ds}{dt}$Here r is a constant. But d $\theta$/dt is the angular velocity $\omega$ and ds/dt is the linear speed v so angular velocity $\omega$ is given by
$\omega$ = $\frac{v}{r}$
or
v = $\omega$ r
Hence the linear speed v at any point on the rotating body is proportional both to the angular speed $\omega$ and the distance from the axis of rotation r.

Linear Speed and Angular Speed

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Lets see how the linear speed and angular speed related to each other using their relations:
If s tells about the linear displacement and t is time taken then Linear speed is
v = $\frac{s}{t}$

If $\theta$ is the angle swept in time t then angular speed ($\omega$) is given as
$\omega$ = $\frac{\theta}{t}$ = $\frac{Central\ Angle}{time}$
Using both the concepts the linear speed is given as
v = $\frac{s}{t}$ = $\frac{r \theta}{t}$ = r $\omega$
Hence v = r $\omega$
It gives the relationship between angular speed and linear speed.

Linear Speed Examples

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Lets see some examples on linear speed:
  1. An athlete running on a straight road. This straight road is nothing but the linear path.
  2. A person traveling moving on the bicycle in the linear path. Its linear speed can be known using bikers pedaling speed, circumference of the wheel and so many perimeters.

How to Find Linear Speed?

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You can determine the linear speed using above formulas based on given problem. Here are two problems on linear displacement you can go through it:

Solved Examples

Question 1: The second hand of the clock is 7 cm as shown in fig. Calculate the linear speed at the tip of the second hand.
Linear Speed Problem

Solution:
 
Given: In one revolution the arc length is
S = 2 $\pi$ r = 2 $\times$ $\pi$ $\times$ 7 = 14 $\pi$ cm
The time it needs for a second hand to travel this distance is t = 1 min = 60 s.

The linear speed of the tip of the second hand is
v = $\frac{s}{t}$
  = $\frac{14 \pi}{60\ s}$
  = 0.733 cm per second
  = 0.00733 m/s.

 

Question 2: A car tire of length 16 inches rotates 180o per second. Calculate its linear speed.
Solution:
 
Given: Radius r = 16 inches, Angular velocity $\omega$ = 180o/sec

The linear speed is given by
v = r $\omega$ = 16 inch $\times$ 180o/sec = 2880 inch/sec

To convert it into miles per hour
v = $\frac{2880 \times 3600}{63360}$ mph
  = $\frac{10368000}{63360}$ mph
  = 163.63 mph.