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

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AS and A Level

IMPORTANT

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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.

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 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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MEDIUM

AS and A Level

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Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 3 - Forces in Two Dimensions>EXERCISE 3D>Q 2

A book of mass $2 kg$ is dragged along a horizontal surface by a rope at $6 °$ above the horizontal. It accelerates at $0.3 m s^{- 2} .$

Draw the force diagram for this situation.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 3 - Forces in Two Dimensions>EXERCISE 3D>Q 2

A book of mass $2 kg$ is dragged along a horizontal surface by a rope at $6 °$ above the horizontal. It accelerates at $0.3 m s^{- 2} .$

Find the tension in the rope and the normal contact force.

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AS and A Level

IMPORTANT

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

A box of mass

10 kg

is pulled along a horizontal surface by a rope with tension

20 N

at an angle

θ

above the horizontal. The box accelerates at

1.2 m s^{- 2} .

Find

θ

and the normal contact force.

HARD

AS and A Level

IMPORTANT

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

A car of mass $1000 kg$ is being towed by two people holding ropes. One pulls with a tension of $80 N$ at an angle of $18 °$ to the direction of motion. The other pulls at an angle of $25 °$ to the direction of motion, as shown in the diagram. Find the tension, $T,$ in the second rope and the acceleration of the car.

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AS and A Level

IMPORTANT

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

A box of mass

20 kg

is dragged by a force of

40 N

at an angle of

15 °

to the direction of motion, and a force of

30 N

at an angle of

θ

to the direction of motion. Find the value of

θ

and the acceleration of the box.

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.

Important Questions on Forces in Two Dimensions

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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.