Linear And Angular Motion Study Material

Linear And Angular Motion Study Material

MOVEMENT or DISPLACEMENT

This is the distance travelled by an object and is usually denoted by x or s. Units of

distance are metres

VELOCITY

This is the distance moved per second or the rate of change of distance with time. Velocity is movement in a known direction so it is a vector quantity. The symbol is v or u and it may be expressed in calculus terms as the first derivative of distance with respect to time so that v = dx/dt or  = ds/dt. The units of velocity are m/s.

SPEED

This is the same as velocity except that the direction is not known and it is not a vector and cannot be drawn as such.

AVERAGE SPEED OR VELOCITY

When a journey is undertaken in which the body speeds up and slows down, the average

velocity is defined as TOTAL DISTANCE MOVED/TIME TAKEN.

Average velocity/speed of a moving object can be calculated as

v = s / t                            (1a)

where

v = velocity or speed (m/s, ft/s)

s = linear distance traveled (m, ft)

t = time (s)

• distance is the length of the path a body follows in moving from one point to another - displacement is the straight line distance between the initial and final positions of the body
• we use velocity and speed interchangeable - but be aware that speed is a measure of how fast or slow a distance is covered, the rate at which distance is covered - velocity is a vector, specifying how fast or slow a distance is covered and the direction

Velocity can be expressed as (velocity is variable)

v = ds / dt    (1f)

where

ds = change in distance (m, ft)

dt = change in time (s)

 ACCELERATION

When a body slows down or speeds up, the velocity changes and acceleration or deceleration occurs. Acceleration is the rate of change of velocity and is denoted with a.

In calculus terms it is the first derivative of velocity with time and the second derivative of distance with time such that a = dv /dt = d2s/dt^2. The units are m/s^2.

Note that all bodies falling freely under the action of gravity experience a downwards

acceleration of 9.81 m/s^2.

Acceleration can be expressed as

a = dv / dt                                   (1g)

where

dv = change in velocity (m/s, ft/s)

WORKED EXAMPLE No.1

A vehicle accelerates from 2 m/s to 26 m/s in 12 seconds. Determine the acceleration. Also find the average velocity and distance travelled.

SOLUTION

a = Δv /t = (26 - 2)/12 = 2 m/s^2

Average velocity = (26 + 2)/2 = 14 m/s

Distance travelled 14 x 12 = 168 m

DISTANCE - TIME GRAPHS

Graph A shows constant distance at all times so the body must be stationary.

Graph B shows that every second, the distance from start increases by the same amount so the body must be travelling at a constant velocity.

Graph C shows that every passing second, the distance travelled is greater than the one before so the body must be accelerating.

VELOCITY - TIME GRAPHS

Graph B shows that the velocity is the same at all moments in time so the body must be travelling at constant velocity.

Graph C shows that for every passing second, the velocity increases by the same amount so the body must be accelerating at a constant rate.

SIGNIFICANCE OF THE AREA UNDER A VELOCITY-TIME GRAPH

The average velocity is the average height of the v-t graph. By definition, the average height of a graph is the area under it divided by the base length

Average velocity = total distance travelled/time taken

Average velocity = area under graph/base length

Since the base length is also the time taken it follows that the area under the graph is the

distance travelled. This is true what ever the shape of the graph. When working out the

areas, the true scales on the graphs axis are used.

WORKED EXAMPLE No.2

Find the average velocity and distance travelled for the journey depicted on the graph

above. Also find the acceleration over the first part of the journey.

SOLUTION

Total Area under graph = A + B + C

Area A = 5x7/2 = 17.5 (Triangle)

Area B = 5 x 12 = 60 (Rectangle)

Area C = 5x4/2 = 10 (Triangle)

Total area = 17.5 + 60 + 10 = 87.5

The units resulting for the area are m/s x s = m

Distance travelled = 87.5 m

Time taken = 23 s

Average velocity = 87.5/23 = 3.8 m/s

Acceleration over part A = change in velocity/time taken = 5/7 = 0.714 m/s^2.

 LINEAR Acceleration and Velocity Equations

Average Velocity

v_a = (v₁ + v₀) / 2                    

where

v_a = average velocity (m/s

v₀ = initial velocity (m/s

v₁ = final velocity (m/s

Final Velocity

v₁ = v₀ + a t                  

where

a = acceleration (m/s²

t = time taken (s

Distance Traveled

s = (v₀ + v₁) t / 2               

where

s = distance traveled (m)

Alternative:

s = v₀ t + 1/2 a t²                

Acceleration

a = (v₁ - v₀) / t                    

Alternative:

a = (v₁² - v₀²) / (2 s) 
the final velocity
v = (v0^2 + 2 a s)^1/2                   

Example - Accelerating Motorcycle

A motorcycle starts with an initial velocity 0 km/h (0 m/s) and accelerates to 120 km/h (33.3 m/s) in 5 s.  

The average velocity can be calculated with eq. (1) to

v_a = ((33.3 m/s) - (0 m/s)) / 2

   =16.7 m\s

The distance traveled can be calculated with eq. (3) to

s = ((0 m/s) + (33.3 m/s)) (5 s) / 2

=166.5 m

The acceleration can be calculated with eq. (4) to

a = ( (33.3 m/s) -  (0 m/s)) / (5 s)

  =6.7 m/s²

ANGULAR MOTION

ANGLE θ
Angle has no units since it a ratio of arc length to radius. We use the names revolution,
degree and radian. Engineers use radian. Consider the arc shown.
The length of the arc is r θ and the radius is r .
The angle is the ratio of the arc length to the radius.
θ = arc length/ radius hence it has no units but it is called radians.
If the arc length is one radius, the angle is one radian so a radian is
defined as the angle which produces an arc length of one radius 

ANGULAR VELOCITY ω

Angular velocity is the rate of change of angle per second. Although rev/s is commonly
used to measure angular velocity, we should use radians/s (symbol ω). Note that since a
circle (or revolution) is 2π radian we convert rev/s into rad/s by ω= 2πN.
Also note that since one revolution is 2π radian and 360o we convert degrees into radian
as follows. θ radian = degrees x 2π/360 = degrees x π/180

DEFINITION
angular velocity = ω = angle rotated/time taken = θ/t

EXAMPLE No.3
A wheel rotates 200o in 4 seconds. Calculate the following.
i. The angle turned in radians?
ii. The angular velocity in rad/s
SOLUTION
θ = (200/180)π = 3.49 rad.
ω = 3.49/4 = 0.873 rad/s

Angular velocity can be expressed as (angular velocity = constant):

ω = θ / t (2)

where

ω = angular velocity (rad/s)

θ = angular distance (rad)

t = time (s)

Angular velocity and rpm:

ω = 2 π n / 60 (2a)

where

n = revolutions per minute (rpm)

π = 3.14...

ANGULAR ACCELERATION α

Angular acceleration (symbol α) occurs when a wheel speeds up or slows down. It is
defined as the rate of change of angular velocity. If the wheel changes its velocity by Δω
in t seconds, the acceleration is
α = Δω / t   rad/s²

 EXAMPLE No.4
A disc is spinning at 2 rad/s and it is uniformly accelerated to 6 rad/s in 3 seconds.
Calculate the angular acceleration.
SOLUTION
α = Δω/t = (ω2 - ω1)/t = (6 -2)/3 = 1.33 rad/s²

Angular velocity can also be expressed as (angular acceleration = constant):

ω = ωo + α t (2c)

where

ω_o = angular velocity at time zero (rad/s)

α = angular acceleration or deceleration (rad/s²)

Angular Displacement

Angular distance can be expressed as (angular acceleration is constant):

θ = ωo t + 1/2 α t²     (2d)

Combining 2a and 2c:

ω = (ωo^2 + 2 α θ)^1/2

Angular Acceleration

Angular acceleration can be expressed as:

α = dω / dt = d2θ / dt²                                    (2e) 

where

dθ = change of angular distance (rad)

dt = change in time (s)

LINK BETWEEN ANGULAR AND LINEAR MOTION

Consider a point moving on a circular path as shown.

The length of the arc = s metres.

Angle of the arc is θ radians

The link is s = r θ

Suppose the point P travels the length of the arc in tme t seconds. The wheel rotates θ

radians and the point travels a distance of r θ.

The velocity along the circular path is v  = r θ/t =r  ω

Next suppose that the point accelerates from angular velocity ω1 to ω2.

The velocity along the curve also changes from v1 to v2.

Angular acceleration = α = (ω2 - ω1)/t

Substituting ω = v /r 

α = (v 2/r  - v 1/r )/t = a/r  hence a =r  α

 EXAMPLE No.5

A car travels around a circular track of radius 40 m at a velocity of 8 m/s. Calculate

its angular velocity.

SOLUTION

v = ωr

ω = v/r = 8/40 = 0.2 rad/s

The tangential velocity of a point in angular velocity - in metric or imperial units like m/s or ft/s - can be calculated as

v = ω r     (2b)

where

v = tangential velocity (m/s, ft/s, in/s)

r = distance from center to the point (m, ft, in)

Angular Moment - or Torque

Angular moment or torque can be expressed as:

T = α I   (2f)

where

T = angular moment or torque (N m)

I = Moment of inertia (lb_m ft², kg m²)