Vectors

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Resolution of vectors

Vectors are often resolved in two components: the vertical component and the horizontal component.

Fv F

Fh

Example: Find the horizontal and vertical and components of the 500 N. y

Fy F = 500 N 40o O X

Horizontal component of F, Fx To find the horizontal component:Step 1: Drop a perpendicular from the 500 N force to the x-axis

Step 2: Calculate the horizontal component as illustrated below:

Fx = cos 40

F Fx = F cos 40 oTherefore, the horizontal component of F, Fx = F cos 40oTo find the vertical component:Step 1: Drop a perpendicular from the 500 N force to the y-axis

Step 2: Calculate the vertical component as illustrated below:

Fy = sin 40 o

F Fy = F sin 40 o

Therefore, the vertical component of F, Fy = F sin 40o

Example: Find the resultant of the four forces shown.

In the horizontal direction to the left, the total force is given by:

(545N x cos 24o) + (425N x cos 15o) (500N x cos 40o) 330N x cos 27o

= 498N + 411N 383N 294N

= 232N

In the vertical upwards direction, the total force is given by:

(545N x sin 24o) - (425N x sin 15o) + (500N x sin 40o) (330N x sin 27o)

= 222N + 321N 110N 150N

= 283N

Using Pythagorass theorem,

F2 = 2322 + 2832

F = 366 N

and tan ( = 232/283

( = 39.3o

The resultant force F therefore has a magnitude of 366 N inclined in a direction 39o to the left of vertically upwards.

Questions

1.

2. Otug A

(

3. A body of mass 10 kg is sliding down a slope inclined at 30o to the horizontal at a constant speed as shown below. Taking g as 10 m/s2, calculate

a. the component of the weight along the inclined plane, hence the force needed to stop the object from sliding,

b. the reaction force R.

4.A microphone of mass 500 g is hung from the ceiling by a wire.

a. What is the tension in the wire?

b. A horizontal force F pulls the wire to make an angle of 30o with the vertical as shown in the figure below. Calculate (i) the new tension in the wire,

(ii) (ii) the magnitude of F. Figure shows how in two dimensions vector F is split into its two components Fv and Fh.

The two components, taken together, have exactly the same effect as the single vector F.

The magnitude of the two components are given by

Fh = F cos (

Fv = F sin (

You should use either the components or the vector itself. All three must not be used together.

F283 N

(

232N

500 N

545N

24o 40o

425N 15o 27o

330N

A lamp is suspended from two wires as shown in the diagram. The tension in each wire is 4.5 N

a. Calculate the magnitude of the resultant force exerted on the lamp by the wires.

b. What is the weight of the lamp? Explain your answer.

4.5 N 4.5 N

40o 40o

Two tugs A and B pull a ship a long the direction XO. Tug A exerts a force on the ship of 3.0 x 104 N at an angle of 15o to XO. Tug B pulls with a force of 1.8 x 104N at and angle ( to XO.

a. Find the angle ( for the resultant

force on the boat to be along XO;

(Hint: when the resultant force is along XO,

the component of the resultant force in the

direction perpendicular to XO is zero.)

b. Find the value of this resultant force.

tug B

30 kN

18 kN

15o

X

R

mg

Component of the weight along the inclined plane

30o

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