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 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.

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 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

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 $0.3 m s^{- 2} .$

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.

HARD

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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HARD

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.

MEDIUM

AS and A Level

IMPORTANT

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

A truck of mass $15000 kg$ is being towed by two ropes. One pulls with a tension of $3000 N$ at an angle of $20 °$ to the direction of motion. The other pulls with a tension, $T,$ at an angle of $10 °$ to the direction of motion. There is resistance of $500 N$ against the motion, in the same line as the motion.

Draw the force diagram for this situation.

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.

Important Questions on Forces in Two Dimensions

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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.