Coplanar forces of magnitudes 58 N , 31 N and 26 N act at a point in the directions shown in the diagram. Given that tan α = 5 12 , find the magnitude and direction of the resultant of the three forces.

Given, tan α = 5 12 By the use of basic trigonometric sin α = 5 13 cos α = 12 13 The components of forces are shown in the following diagram The vertical component F y = 58 - 26 sin α F y = 58 - 26 × 5 13 = 48 N The Horizontal component is, F x = 26 cos α + 31 F x = 26 × 12 13 + 31 = 55 N So the magnitude of the resultant force is F = F 2 x + F 2 y F = 48 2 + 55 2 = 73 N Let the direction of the resultant force is θ tan θ = 48 55 ⇒ θ = 41 . 1 ° , from positive x direction.

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Three forces act on a particle in equilibrium in the horizontal plane, as shown in the diagram. By resolving in a direction perpendicular to $F,$ show that $θ = 47.2 °$ and find $F .$

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>END-OF-CHAPTER REVIEW EXERCISE 3>Q 4

Two people drag a car of mass

1200 kg

forward with ropes. One pulls with force

400 N

on a bearing of

005 ° .

One pulls with force

360 N

on a bearing of

352 ° .

Find magnitude of the acceleration and its direction to the nearest

0.1 ° .

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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>END-OF-CHAPTER REVIEW EXERCISE 3>Q 5

A boat is in equilibrium held by a rope to the shore. The rope exerts a force $T$ at an angle $θ$ from north. The wind blows the boat with force $40 N$ in a northwest direction. The current pushes it south with a force of $50 N .$ Show that $T \sin θ = 20 \sqrt{2}$ and find an expression for $T \cos θ$ . Hence, show that $\tan θ = \frac{8 + 10 \sqrt{2}}{17}$ and find $θ$ and $T .$

Question Image

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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>END-OF-CHAPTER REVIEW EXERCISE 3>Q 6

A car of mass

300 kg

is on a slope, which is at an angle of

5 °

to the horizontal. When it is pulled down the slope by a rope parallel to the slope with a force of

T,

it accelerates at

2 m s^{- 2} .

Find the acceleration of the car when it is pulled up the slope by a rope parallel to the slope with a force of

T .

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AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 3 - Forces in Two Dimensions>END-OF-CHAPTER REVIEW EXERCISE 3>Q 8

A girl can drag a stone block of mass

18 kg

up a slope at an angle of

13 °

to the horizontal with an acceleration of

0.7 m s^{- 2} .

Assuming this is the maximum force she can exert to drag the block, find the mass of the heaviest stone block she would be able to drag up the slope.

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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>END-OF-CHAPTER REVIEW EXERCISE 3>Q 13

A ball of mass

m kg

slides down a slope, which is at an angle of

θ °

to the horizontal. It passes two light gates

x m

apart. At the first gate, the speed of the ball is measured as

u m s^{- 1},

and at the second its speed is measured as

v m s^{- 1} .

Assuming the resistance is constant, show the resistance force has a total size of

\frac{m}{2 x} (2 xgsin θ + u^{2} - v^{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>END-OF-CHAPTER REVIEW EXERCISE 3>Q 14

A car of mass

m kg

is rolling down a slope of length

x m,

which is at an angle of

30 °

to the horizontal. It has a booster that provides a force of

m g N

over a distance of

1 m

, which the driver sets off at a distance

s m,

after the car starts moving. Assuming the booster is used before the end of the slope, show that the speed at the bottom of the slope is given by

v^{2} = g (x + 2)

and deduce that the final speed is independent of when the booster is applied. (Note that if the booster were applied for a fixed time rather than a fixed distance this would not be true.)

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AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 3 - Forces in Two Dimensions>END-OF-CHAPTER REVIEW EXERCISE 3>Q 15

Question Image

Coplanar forces of magnitudes $58 N, 31 N$ and $26 N$ act at a point in the directions shown in the diagram. Given that $\tan α = \frac{5}{12},$ find the magnitude and direction of the resultant of the three forces.

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AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Mechanics Course Book>Chapter 3 - Forces in Two Dimensions>END-OF-CHAPTER REVIEW EXERCISE 3>Q 16

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A particle $P$ of mass $1.05 kg$ is attached to one end of each of two light inextensible strings, of lengths $2.6 m$ and $1.25 \mathrm{~m}$. The other ends of the strings are attached to fixed points $A$ and $B$, which are at the same horizontal level. $P$ hangs in equilibrium at a point $1 \mathrm{~m}$ below the level of $A$ and $B$ (see diagram). Find the tensions in the strings.

Three forces act on a particle in equilibrium in the horizontal plane, as shown in the diagram. By resolving in a direction perpendicular to F, show that θ=47.2° and find F.

Important Questions on Forces in Two Dimensions

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Three forces act on a particle in equilibrium in the horizontal plane, as shown in the diagram. By resolving in a direction perpendicular to $F,$ show that $θ = 47.2 °$ and find $F .$