A wooden block is held in position by three horizontal forces, as shown in the diagram. One acts to the left with force 56 N . One acts with force F at an angle θ . where sin θ = 3 5 , , above the rightwards horizontal. One acts with force G at an angle φ , where sin φ = 15 17 , , below the rightwards horizontal. Find F and G .

Given Two forces F . G acting on a block making angles θ , φ with the horizontal as shown, to the right side A force of 56 N is acting in the opposite direction. Given that sin θ = 3 5 , sin φ = 15 17 so we have, cos θ = 4 5 and cos φ = 8 17 . At equilibrium, the net force on the block is zero. Equating the vertical forces, we get, F sin θ = G sin φ or F × 3 4 = G × 8 17 or F = 32 21 G .....................................(1) Equating the horizontal forces, we have, F cos θ + G cos φ = 56 N Substituting the value in this equation, we have, 32 21 G × 4 5 + G × 8 17 = 56 or 128 105 G + 8 17 G = 56 Or 1 . 22 G + 0 . 47 G = 56 or G 33 . 13 N ...............................(1) And F = 33 . 13 × 32 21 = 50 . 5 N .................................................(2)

A block with weight 44 N is held in equilibrium by two ropes, one with tension, T 1 , acting at angle sin - 1 15 17 to the upwards vertical and the other with tension, T 2 , acting at angle sin - 1 5 13 to the upwards vertical. Find T 1 and T 2 .

Given the mass of the block= 44 N . sin α = 15 17 , sin β = 5 13 so we have cos α = 8 17 , cos β = 12 13 . The following forces are acting on the block. (1) the weight of the block 44 N directed vertically downwards, (2) The horizontal component if the tension T 1 = T 1 sin α acting towards the left side. (3) The horizontal component of the tension T 2 = T 2 sin β acting towards the right side . (4) and (5) The vertical components of the tensions T 1 cos α and T 2 cos β , acting vertically upwards, At equilibrium, Equating the horizontal forces, we have, T 1 sinα = T 2 sinβ or T 1 × 15 17 = T 2 × 5 13 or T 1 = 0 . 43 T 2 ..........(1) Equating the vertical forces, we have, T 1 cos α + T 2 cos β = 44 N or 0 . 43 × 8 17 T 2 + 12 13 T 2 = 44 Or 0 . 2 T 2 + 0 . 92 T 2 = 44 or T 2 = 39 . 2 N ......................................(2) Substituting this value in Eq.(1), we get T 1 = 0 . 43 × 39 . 2 = 16 . 9 N .....(3)

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A wooden block of weight $20 N$ is at rest on a horizontal surface. It is pulled by a force of $30 N$ acting at $10 °$ above the horizontal, as shown in the diagram, and remains at rest because of a horizontal frictional force, $F .$

Find the size of $F$ and the size of the normal contact force.

Important Questions on Forces in Two Dimensions

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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 3A>Q 7

A winch is dragging a caravan along a horizontal road at constant velocity. The caravan has mass $750 kg$ . The winch provides a force of $850 N$ and acts at angle $θ$ above the horizontal, as shown in the diagram. There is friction of $700 N .$

Draw the force diagram for this situation.

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AS and A Level

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

A winch is dragging a caravan along a horizontal road at constant velocity. The caravan has mass $750 kg .$ The winch provides a force of $850 N$ and acts at angle $θ$ above the horizontal, as shown in the diagram. There is friction of $700 N .$

Find the value of $θ$ and size of 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 3A>Q 8

A box of weight $50 N$ is being dragged at constant velocity along a horizontal road by a force, $F,$ , acting at $15 °$ above the horizontal. It experiences friction of $10 N .$

Draw the force diagram for this situation.

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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 3A>Q 8

A box of weight $50 N$ is being dragged at constant velocity along a horizontal road by a force, $F,$ , acting at $15 °$ above the horizontal. It experiences friction of $10 N .$

Find $F$ 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 3A>Q 9

A small aeroplane of mass

5000 kg

is towed along a runway at constant speed by a rope acting at

20 °

below the horizontal. There is friction and air resistance horizontally with total force

4000 N .

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 3A>Q 10

A wooden block is held in position by three horizontal forces, as shown in the diagram. One acts to the left with force $56 N .$ One acts with force $F$ at an angle $θ .$ where $\sin θ = \frac{3}{5},$ , above the rightwards horizontal. One acts with force $G$ at an angle $φ,$ where $\sin φ = \frac{15}{17},$ , below the rightwards horizontal. Find $F$ and $G$ .

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AS and A Level

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Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 3 - Forces in Two Dimensions>EXERCISE 3A>Q 11

A block with weight

44 N

is held in equilibrium by two ropes, one with tension,

T_{1},

acting at angle

\sin^{- 1} \frac{15}{17}

to the upwards vertical and the other with tension,

T_{2},

acting at angle

\sin^{- 1} \frac{5}{13}

to the upwards vertical. Find

T_{1}

and

T_{2} .

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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 3A>Q 12

A box with weight

400 N

is at rest on a horizontal surface. A man is pulling on a rope to try to get the box to move. The force he can exert depends on the angle at which he holds the rope, so that when the rope is at an angle

1600 \sin θ N

above the horizontal, the force he exerts is

1600 \sin θ N .

He starts by holding the rope horizontally and gradually increases the angle, thereby increasing the force. Another man tries to prevent this motion of the box, by pulling horizontally. He can exert a maximum force of

700 N .

Find the angle at which the box can no longer remain on the ground. Hence, determine whether the box lifts off the ground first or slides along the ground first.

A wooden block of weight 20 N is at rest on a horizontal surface. It is pulled by a force of 30 N acting at 10° above the horizontal, as shown in the diagram, and remains at rest because of a horizontal frictional force, F. Find the size of $F$ and the size of the normal contact force.

Important Questions on Forces in Two Dimensions

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A wooden block of weight $20 N$ is at rest on a horizontal surface. It is pulled by a force of $30 N$ acting at $10 °$ above the horizontal, as shown in the diagram, and remains at rest because of a horizontal frictional force, $F .$

Find the size of $F$ and the size of the normal contact force.