Vectors and Forces

Refresher

In the following diagram, Vector a  has direction θ in the (x,y) plane and is shown with  unit vectors i and j.

If a represents the position vector of the point A(a_x, a_y)

then  a = a_xi + a_yj

 

Comparing coefficients of point A

Further

 

For calculations with vertical and horizontal components, avoid confusion by always using the angle between the vector and the horizontal.

 

If P is a particle, the position vector of P at time t is given r_P

The velocity vector of P is

otherwise written

 

The acceleration of P at time t is

also written as

 

Example

A particle moves in the x-y plane relative to a fixed point O.

The particle is initially located at the point -2i + 3j, where i and j  are unit vectors in the directions of the x- and y-axis respectively.

t seconds after the start of its motion, the velocity of the particle is given by v = 2sin ti +3cos5tj

Find expressions for the acceleration and position of the particle t seconds after the start of its motion.

 

Solution

Acceleration

 

Displacement

 

Example

A particle moves in the x-y plane relative to a fixed point O.

The particle is initially located at the point -2i + 3j, where i and j  are unit vectors in the directions of the x- and y-axis respectively and has velocity v = 3i +4j m/s and is accelerating at 2i +4j m/s²

What is the velocity, speed and position of of the particle 5 seconds after the start of its motion?

 

When t = 5

V=13i +24j m/s

When t = 5

When t = 5

r = 38i +73j m

 

Effects of wind

Example

A ship travelling westward at 8 km/h is subjected to a wind blowing from the north at 60 km/h

 

 

What effect does the wind have on the ship ?

 

The ship is blown offcourse by 82.4 degrees.

 

 

 

 

Principle of Two Forces

If an object is at rest and remains so, and is acted upon by just two forces, then

a) the forces are equal in magnitude and opposite in direction.

b) both forces act along the line joining their points of application.

 

 

Law of Statics

If an object acted upon by various forces is at rest and remains so, then the vector sum of all of the forces acting  upon is equal to zero.

Example

A baby is suspended from two springs in a state of static equilibrium as shown. What is the mass (M Kg) of the baby ?

Take the acceleration due to gravity to be 9.81 m/s²

Split into components:

Looking at the vertical components,

 

 

Alternatively, using Lami's Theorem

 

 

 

Frame of Reference

A frame of reference is used to fix the point of the origin of a coordinate system that can then be used to take measurements for calculations within that frame.

Example

Here, Alfie is sat stationary, watching Bert run at a constant speed after Paul, who is driving away at a constant speed.

From Alfie's point of view, at any given time the co-ordinate of Bert is x_BA and the co-ordinate of Paul is x_PA

 

From Bert's point of view, at any given time the co-ordinate of Paul is x_PB

so

x_PA = X_PB + X_BA

 

 

Relative position, velocity and acceleration

The positions of two bodies, A and B are shown from the origin, O.

The instantaneous position of A can be presented by the vector r_a and that of B as r_b

 

 

 

Relative Motion

When two frames of reference , A and B , are moving relative to each other at a constant velocity, the velocity of a particle P as measured by an observer in frame A is

Which can be written without the overhead vector arrows for motion in a single axis

V_PA = V_PB +V_BA

Re-arranging gives

some texts use

_BV_A = V_PB - V_PA

to show the velocity of B relative to A

To find the acceleration of P as measured from A and B, the velocity vectors are differentiated with respect to time ;

Since V_BA is constant, the last term is zero

a_PA = a_PB

Observers on different frames of reference that move at constant velocity relative to each other will measure the same acceleration for a moving particle.

 

Example

Ship A is moving with speed 10 kilometres per hour due west and ship B is moving with speed 8 kilometres per hour due north.
Find the magnitude and direction of the velocity of ship A relative to ship B.

 

The question wants the velocity of ship A relative to ship B

_AV_B = V_PA - V_PB

so

_AV_B = V_A - V_B

 

The direction of B is reversed for the resulting velocity diagram.

 

 

The bearing of ship A from ship B is 270°-38.7°≈231°

This can also be written as W39°S, read 39 degrees south from west or S51°W , read 351 degrees west of south.

The magnitude of velocity of ship A from ship B is 12.81 km/h