Four forces on an object, A , B , C and D , result in no net force. If the angle between forces A and B is α and the angle between forces C and D is γ , show that A 2 + B 2 + 2 ABcosα = C 2 + D 2 + 2 C D cos γ .

We simply draw the vector diagram of all given forces as given below Now, we use the Cosine formula as follows c 2 = a 2 + b 2 - 2 a b cos C So, d 2 = A 2 + B 2 - 2 A B cos 180 ° - α Similarly d 2 = C 2 + D 2 - 2 C D cos 180 ° - γ Now we compare both of this equation A 2 + B 2 - 2 A B cos 180 ° - α = C 2 + D 2 - 2 C D cos 180 ° - γ = d 2 But, cos 180 ° - α = - cos α cos 180 ° - γ = - cos γ So that A 2 + B 2 + 2 A B cos α = C 2 + D 2 + 2 C D cos γ

HARD

AS and A Level

IMPORTANT

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A mass of $5 kg$ is held in equilibrium by two ropes with tensions of $30 N$ and $40 N .$ Find the angles that the ropes make with the vertical.

Important Questions on Forces in Two Dimensions

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 3 - Forces in Two Dimensions>EXERCISE 3C>Q 3

A mass of

7 kg

is held in equilibrium by two ropes. One has tension

20 N

and acts at

40 °

to the upwards vertical. Find the tension in the other rope and the angle that it makes with the upwards vertical.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 3 - Forces in Two Dimensions>EXERCISE 3C>Q 4

A ship is held in place by two ropes with forces $40 N$ and $35 N,$ as shown in the diagram, which prevent the wind blowing it away. The wind has force $F$ and acts at an angle $θ$ to the $35 N$ force, as shown. Find the sizes of $θ$ and $F .$

Question Image

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 3 - Forces in Two Dimensions>EXERCISE 3C>Q 5

Three ropes pull a boat, which remains in equilibrium. The ropes act due north and on bearings of

100 °

and

210 ° .

The one acting north has tension

25 N .

Find the tensions in the other ropes.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 3 - Forces in Two Dimensions>EXERCISE 3C>Q 6

A box has two ropes holding it in place. It is pushed by a force of

10 N .

The angles between the force and the ropes are

120 °

and

150 ° .

Find the tensions in the ropes.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 3 - Forces in Two Dimensions>EXERCISE 3C>Q 7

An $8 N$ force, a $9 N$ force and a $10 N$ force on an object result in no net force. Find the angle between the $8 N$ and the $9 N$ forces.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 3 - Forces in Two Dimensions>EXERCISE 3C>Q 9

A particle is held in equilibrium by three forces. Two of the forces have the same size,

F N .

Prove that the third force acts along the line of the angle bisector of the lines of action of the other two forces.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 3 - Forces in Two Dimensions>EXERCISE 3C>Q 10

Four forces on an object,

A, B, C

and

D,

result in no net force. If the angle between forces

A

and

B

α

and the angle between forces

C

and

D

γ,

show that

A^{2} + B^{2} + 2 ABcosα = C^{2} + D^{2} + 2 C D \cos γ .

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 3 - Forces in Two Dimensions>EXERCISE 3D>Q 1

A wooden block of mass $5 kg$ Is on a horizontal surface. It is dragged by a force of $20 N$ Acting at $14 °$ Above the horizontal, as shown in the following diagram. Find the acceleration of the block and the normal contact force.

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A mass of 5 kg is held in equilibrium by two ropes with tensions of 30 N and 40 N. Find the angles that the ropes make with the vertical.

Important Questions on Forces in Two Dimensions

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A mass of $5 kg$ is held in equilibrium by two ropes with tensions of $30 N$ and $40 N .$ Find the angles that the ropes make with the vertical.