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Middle Years

13.06 Applications of Vectors

Worksheet
Applications of vectors
1

A soccer ball is kicked with a velocity of 55 \text{ km/h} and an angle of inclination of 50 \degree.

a

Find the horizontal component of the velocity to the nearest \text{km/h} .

b

Find the vertical component of the velocity to the nearest \text{km/h} .

2

A force has a magnitude of 78 \text { N} and makes an angle of 27 \degree counterclockwise from the horizontal.

a

Find the horizontal component of the force. Round your answer to the nearest tenth of a Newton.

b

Find the vertical component of the force. Round your answer to the nearest tenth of a Newton.

3

A force has a magnitude of 37 \text { N} and makes an angle of 48 \degree clockwise from the vertical.

a

Find the horizontal component of the force.Give your answer to the nearest tenth of a Newton.

b

Find the vertical component of the force.Give your answer to the nearest tenth of a Newton.

4

Express each of the following forces as a vector in terms of \mathbf{i} and \mathbf{j}:

a

240 \text { N} applied at 60 \degree to the horizontal.

b

270 \text { N} applied at 30 \degree to the vertical.

5

Express each of the following forces as a vector in component form:

a

270\text { N} that is applied downwards.

b

160\text { N} that is applied to the east.

6

Let forces \overrightarrow{A} and \overrightarrow{B} be as shown in the following diagrams. If C newtons \text { (N)} is the magnitude of the resultant force, calculate C correct to the nearest whole number.

a
b
7

Two forces, of magnitudes 30 newtons and 26 newtons, are acting on the same object separated by an angle of 140 \degree. If F newtons is the magnitude of the resultant force, calculate F correct to one decimal place.

8

A 150 \text { N} force is to be resolved into two rectangular components such that the magnitude of one component is 2 times the magnitude of the other.

a

Find x, the exact magnitude of the smaller component.

b

Find \theta, the angle between the smaller component and the 150\text { N} force. Round your answer to the nearest degree.

9

If a 12 \text { N} force was resolved into two equal rectangular components, find x, the magnitude of each component.

10

A boat heads directly across a river with a velocity of 13 \text{ m/s}. If the river flows at 4\text{ m/s} in the opposite direction, find the boat's resultant velocity.

11

A box is at rest on a ramp that is inclined at an angle of 12 \degree. It is being held at rest by a 2134 \text { N} force perpendicular to the ramp and a 454 \text { N} force of friction parallel to the surface of the ramp. Find the weight of the box to the nearest tenth of a newton.

12

A pair of wet jeans is hanging from the middle of a clothes line. The pair of jeans weighs 40 \text { N} and the ends of the clothes line are at 16 \degree below the horizontal.

a

Find \left|\mathbf{T} \right|, the magnitude of the tension in each half of the clothesline. Round your answer to the nearest Newton.

b

What are the directions of the tension vectors on each half of the clothes line?

13

A chandelier with a weight of 310 \text { N} is hanging from two chains nailed to the ceiling at angles of 61\degree and 68 \degree respectively.

a

Find \left|\mathbf{T_{1}}\right|, the magnitude of the tension in the chain that is at 61 \degree to the horizontal. Round your answer to the nearest Newton.

b

Find \left|\mathbf{T_{2}}\right|, the magnitude of the tension in the chain that is at 61 \degree to the horizontal. Round your answer to the nearest Newton.

14

A decoration weighing 40 \text { N} is hanging from two ropes of equal length that are attached to the ceiling. If the angle between the ropes is 140 \degree, find \left|\mathbf{T}\right|, the magnitude of the tension in each rope. Round your answer to the nearest newton.

15

A wooden crate weighing 1500 \text { N} is hanging from three 8 \text { m} chains attached to the ceiling of a warehouse at three points that form an equilateral triangle with a side length of 12 \text { m}. Find \left|\mathbf{T}\right|, the tension in each chain.

16

An object weighing 75 \text { N} is on a ramp that is inclined at 17 \degree. It is held at rest by two rectangular components: a force \mathbf{P} perpendicular to the ramp and a friction force \mathbf{F} parallel to the ramp.

a

Find the magnitude of the force perpendicular to the ramp. Round your answer to the nearest tenth of a Newton.

b

Find the magnitude of the force of friction parallel to the ramp. Round your answer to the nearest tenth of a Newton.

17

A car weighing 19\,000 \text { N} is kept from rolling down an inclined road by a force applied at an angle of 7 \degree to the road. If the road is at an angle of 7 \degree to the horizontal, what is the magnitude of the force? Round your answer to the nearest Newton.

18

Find the acceleration of a 40 \text { kg} object as it slides down a ramp inclined at 11 \degree to the horizontal if there is no friction. Assume the acceleration due to gravity is 9.8 \text{ m/s}^2and Round your answer to one decimal place.

19

A car travelling north collides with a car travelling east. The resultant momentum after the collision is 36\,000 \text{ kg} \bullet \text{m/s N } 60 \degree \text{E} .

a

Find the momentum of the car travelling north before the collision.

b

Find the momentum of the car travelling east before the collision.

20

A ship is 30 \text { km} at a bearing of \text{N }50.7 \degree \text{E} from a lighthouse.

a

Find the horizontal component of the ship's displacement. Round your answer to the nearest tenth of a kilometre.

b

Find the vertical component of the ship's displacement. Round your answer to the nearest tenth of a kilometre.

21

A 17 \text { kg} photo frame is hung on a wall with a string attached to its top-left and top-right corners. The middle of the string is placed on a nail on the wall, and the angle between each half of the string and the horizontal is 45 \degree. Find \left|\mathbf{T}\right|, the magnitude of the tension in each half of the string, to the nearest tenth of a Newton. Assume the acceleration due to gravity is 9.8 \text{ m/s}^2.

22

Mahmoud is pulling a 90 \text { kg} esky with a rope that is at 30 \degree with the ground. The force he is applying has a horizontal component of 112 \text { N}. What is the total force that he is applying? Round your answer to the nearest Newton.

23

A 220 \text { N} force is resolved into the components 120\text { N} and 160\text { N}. Are these components rectangular?

24

A lamp with a mass of 4 \text { kg} is hanging from a rope attached to the ceiling.

a

Is the direction of the tension acting on the lamp up or down?

b

What is the magnitude of the tension? Assume the acceleration due to gravity is 9.8 \text{ m/s}^2.

25

An object with a mass of 7\text { kg} is on a ramp that is inclined at 27 \degree. It is held at rest by two rectangular components: a force \mathbf{P} perpendicular to the ramp and a friction force \mathbf{F} parallel to the ramp. Assume the acceleration due to gravity is 9.8 \text{ m/s}^2.

a

Calculate the weight force of the object.

b

Find the magnitude of the force perpendicular to the ramp. Give your answer to the nearest tenth of a Newton.

c

Find the magnitude of the force of friction parallel to the ramp. Round your answer to the nearest tenth of a Newton.

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