Resolution Of Vectors (student Worksheet) p251z

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

Fv

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.

Φ

The magnitude of the two components are given by Fh = F cos φ Fv = F sin φ

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 o Therefore, the horizontal component of F, Fx = F cos 40o To 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 1

Example: Find the resultant of the four forces shown.

F

500 N 545N

24o 425N 15o

283 N θ

40o 27o

330N 232N

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 Pythagoras’s theorem, F2 = 2322 + 2832 F = 366 N and tan θ = 232/283 θ = 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. 4.5 N

4.5 N 40o 40o

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.

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2. O tug B

tug A 30 kN

θ 15

o

18 kN

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 θ to XO. a. Find the angle θ 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.

X

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. b.

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

Component of the weight along the inclined plane mg

30o

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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.

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