A trolley on the tracks is pulled horizontally with a force of 600 N. The applied force acts at an angle with respect to the direction of the tracks. What component of the force acts in the direction of motion of the trolley?

Example 2.

Jack pushes a 30 kg box with a force of 300 N at an angle downwards as shown in the figure. Graphically determine the magnitude of the force component in the direction of motion of the box and the force component directed vertically downwards. With what force does the box act on the ground?

A ball weighing 8 N is suspended on a string. It is displaced by an angle (to point A) and allowed to swing.

Determine graphically, the forces in the string and in the direction of motion at point A.
What are the forces in the string and in the direction of motion when the ball moves through point B?

The drawn bow acts on an arrow with a force of 300 N when it is fired. With what force are the limbs of the bow tensioned if the angle between the arrow and each of the limbs is equal to . Solve the problem mathematically with the help of the knowledge of geometry - equilateral triangles.

A log is placed on a slope with an inclination of , as shown in the figure. The weight of the log is 5 N. With how much force does the log act perpendicular to the slope and with how much force does it act parallel to the slope?

John skis down a slope with an inclination of . His mass is 50 kg. How much force does he apply on the slope while still on the slope?

What are the two forces of John on the slope?
What are the two forces when he skis on the plain surface?

A man strikes a log with an axe with a force of 200 N. With what force does the axe act on both sides of the log? The blade of the axe in the cross section makes an angle of as .

Four perpendicular forces of magnitudes 80 N, 40 N, 60 N, and 20 N act on a ball. Find the fifth force that can act with the four other forces on the ball so that the ball moves uniformly and in a straight line.

A boat is pulled up a stream by two ropes, as shown in the picture. The boat is towed by each rope with a force of 80 N. The angle between the two ropes is . What is the resultant force of both ropes on the boat?

10.

A 60 N box is pulled horizontally by a force of 80 N. Graphically determine the magnitude and direction of the resultant force on the box.

11.

Two forces 8 N and 6 N are perpendicular to each other as shown in the figure. Their resultant force has also been drawn in the picture. Calculate the size of the resultant force.

12.

Three forces of magnitude 4 N each are inclined at to each other, as shown in the figure. What is their resultant force?

13.

The figure below shows three forces in a plane, acting at a point. The forces are drawn on a scale of .

What is the value of each force?
What is the magnitude and direction of the resultant force?

14.

A string is attached to the ceiling at both ends so that it is at an angle on the left-hand side and on the right-hand side, to the ceiling- see the figure below. A 4 N weight is hung on the string. Graphically determine what the forces are in both parts of the string?

15.

On a wall hangs a picture in a frame weighing 2 kg with a string attached to the frame on two sides such that the string encloses an angle of with respect to the upper edge of the frame - see figure. With how much force do both ends of the string hold the image?

16.

Two equal forces are arranged such that their resultant force is equal in magnitude and direction as the two forces. What kind of shape is formed from such arrangement?

17.

A tug pulls a ship along a channel with a force of 700 N. By what forces would you have to tow the ship at angles with respect to the direction of movement of the ship in order for the forces to have the same effect on the ship?

18.

A folding shelf is attached to the wall. At an angle is a rope that holds the shelf in a horizontal position - see picture. The rope can withstand a maximum force of 100 N. If we place a package weighing 5 kg on the shelf, will the rope break?

19.

James stands at the top of a ladder as shown in the picture. His weight is 500 N. With what forces do the load-bearing columns of the ladder act on the base if the angle between the legs of the ladder is ? We ignore the weight of the ladder.

20.

A sphere lies on a flat surface. Two perpendicular forces of magnitude 12 N each begin to act on it.

What is the resultant of both forces?
Which direction will the ball move and is the movement uniform or not?
How much force do we need to add to keep the movement uniform?

21.

A ball weighing 600 N hangs from a crane. Determine the direction and magnitude of the force caused by the weight of the ball on the left and right pillars of the crane structure according to the figure below.

22.

The wood saw stand (goat) has two legs - see picture. The angle between them is like . We place a log weighing 600 N between the legs. With what force does the log act on each of the legs?

23.

A string is attached to the ceiling at one end and to the wall at the other end, as shown in the figure. The angle between the string and the ceiling is , and between the string and the wall is . A weight of 60 N is hung on the string. What is the force in each part of the string?

24.

A 50 m long climbing rope weighs 2.6 kg. Calculate its weight per metre of length.

25.

Two forces of magnitudes 80 N and 60 N act on a body at an angle to each other. What is the magnitude of the third force that can be added to the two forces so that the body moves uniformly and in a straight line?

26.

Calculate the components and absolute value of the resultant of the vectors (forces) given in the x, y coordinate system in the sketch below.

Forces in x, y coordinate system

27.

A ball rests on the table - Figure A. No traction or thrust is applied to it. At some point, three equal forces of magnitude 4 N begin to act on the ball. The angle between each pair of these forces is - Figure B. What will be the motion of the ball after the forces start to act?

28.

A shelf with a brace is attached to the wall as shown in the picture. At the edge of the shelf is a load weighing 100 N. What are the forces on the shelf and the brace?

Attached shelf with bracket

29.

The body slides uniformly and in a straight line at a speed of 3 m/s. At some point, two equal and perpendicular forces of magnitude 4 N each begin to act on it - see figure. How does the speed of motion of the body change? What force should be added to both forces to maintain the speed and direction of motion?

30.

A boat is tied with ropes to the bank of the river as shown in the picture. Due to the flow of the river, the rope at an angle is tensioned with a force of 600 N. With what force does the river act on the boat?

The boat is tied to the bank with two ropes.

31.

A body weighing 6 kg hangs on two ropes attached to the ceiling as shown in the figure. Calculate the forces in both ropes.

Body on two ropes

32.

A weight of 10 N is hung in the middle of a rope. What is the force in the left and right arms of the rope when the angle between the two arms of the rope is:

Weight on a rope

33.

A rope is attached to the ceiling on one side and passed over the attached pulley on the other side. Between the attached end of the rope and the pulley hangs a body of weight , and on the other side of the pulley is a body of weight - see the figure below. The system is in equilibrium. Calculate the angle between the rope and the horizontal axis.

34.

Three bodies are attached to a rope and hung via two attached pulleys as shown in the figure below. The weights of the second and third bodies are 10 N and 18 N respectively.

What is the angle of the right side of the rope with respect to the horizontal?
What must be the weight of the first body for the left side of the rope to be horizontal?

35.

The diagram shows a uniform beam PQ. The length of the beam is and its weight is . The beam is supported on a pivot from end P. A load of weight is hung from end P and the beam is in equilibrium.

What is the value of ?

36.

A uniform metre rule of weight is pivoted at the mark. A load is suspended from one end, causing the rule to rotate about the pivot.

At the instant when the rule is horizontal, what is the resultant moment about the pivot?

37.

Two rigid steel beams XY and YZ are fixed at their lower ends and are hinged at Y. Each beam is inclined at to the horizontal, as shown. A weight of hangs from Y. The structure is in equilibrium.

What is the force exerted by each beam on the hinge at Y?

38.

Three coplanar forces act on a block. Which diagram shows the directions of the forces such that the block could be in equilibrium?

39.

A ladder rests in equilibrium on rough ground against a rough wall.

Its weight W acts through the centre of gravity G. Forces also act on the ladder at P and at Q. These forces are P and Q respectively. Which vector triangle represents the forces on the ladder?

40.

A uniform bar of length L and weight W rests horizontally on two supports X and Y.

Support X exerts a vertical force at a distance of from one end of the bar. Support Y exerts a vertical force at the other end of the bar. The bar is in equilibrium. What is the ratio ?

41.

A uniform metre rule is pivoted at the mark, as shown.

The rule balances when a mass is hung from the mark. What is the mass of the metre rule?

42.

- Define the moment of a force about a point.
- Determine the SI base units of the moment of a force.
A uniform rigid rod of length is shown in the figure below.

The rod has a weight of and is made of wood of density . Calculate the cross-sectional area , in , of the rod.
A fishing rod AB, made from the rod in (b), is shown in the figure below.

End A of the rod rests on the ground and a string is attached to the other end B. A support stick exerts a force perpendicular to the rod at point C. The weight of the rod acts at point D. The tension in the string is in a direction perpendicular to the rod. The rod is in equilibrium and inclined at an angle of to the vertical.
- Show that the component of the weight that is perpendicular to the rod is .
- By taking moments about end A of the rod, calculate the tension .

43.

A student balances a ruler on a fulcrum set at the mark. She then places a mass on the mark and a mass on the mark, as shown.

Which mass should she place on the mark to restore the balance?

44.

State the two conditions that must be satisfied for a body to be in equilibrium.
Three co-planar forces act on a body that is in equilibrium.
- Describe how to draw a vector triangle to represent these forces
- State how the triangle confirms that the forces are in equilibrium.
A weight of hangs vertically by two strings AB and AC, as shown in the figure below

For the weight to be in equilibrium, the tension in string AB is and in string AC it is . On the above figure, draw a vector triangle to determine the magnitudes of and .
By reference to the figure, suggest why the weight could not be supported with the strings AB and AC both horizontal.

45.

An object, made from two equal masses joined by a light rod, falls with uniform speed through air. The rod remains horizontal. Which statement about the equilibrium of the system is correct?

It is not in equilibrium because it is falling steadily.
It is not in equilibrium because it is in motion.
It is not in equilibrium because there is a resultant torque.
It is in equilibrium because there is no resultant force and no resultant torque.

46.

A rigid uniform beam is pivoted horizontally at its midpoint. Different vertical forces are applied to different positions on the beam. In which diagram is the beam in equilibrium?

47.

A box of length and weight is placed on a horizontal table, with the greater part of its length overhanging the edge of the table. The edge of the table acts as a pivot. The centre of gravity of the box is at its geometric centre. To balance the box, a uniform sphere of diameter is placed inside the box, touching one end, as shown.

Assume that the forces acting on the box are in the plane of the diagram. What is the minimum mass of the sphere that is needed to maintain the system in equilibrium?

48.

A uniform beam AB is attached by a frictionless hinge to a vertical wall at end A. The beam is held so that it is horizontal by a metal wire CD, as shown in the figure below:

The beam is of length and weight . A block of weight rests on the beam at a distance of from end B. The wire is attached to the beam at point D which is a distance of from end B. The wire exerts a force on the beam of at an angle of to the horizontal. The beam is in equilibrium.

Calculate the vertical component of the force exerted by the wire on the beam.
By taking moments about A, calculate the weight of the block.
The hinge exerts a force on the beam at end A. Calculate the horizontal component of this force.
The block is now placed closer to point D on the beam. State whether this change will increase, decrease, or have no effect on the tension in the wire.

49.

Forces are applied to a rigid object. The forces all act in the same plane. In which diagram is the object in equilibrium?

50.

Two parts of a sailing boat are the mast and the boom. The mast is a vertical rigid beam and the boom is a horizontal rigid beam. One end of the boom is attached to the mast by a pivot. The other end of the boom is connected to the mast by a rope, as shown.

The rope is at an angle of to the horizontal and exerts a tension force on the boom. The weight of the boom is . The mass of the boom is uniformly distributed along its length. The boom is in equilibrium. What is the magnitude of T?

51.

A rigid rod XY has negligible mass and length . The rod is suspended from a fixed point P by a string attached to end X. An object of mass is suspended by a string that is attached to the rod at a distance of from end X, as shown.

Which vertically upward force acting on end Y of the rod would hold the rod horizontally in equilibrium?

52.

An object is suspended by two ropes. One rope has a tension of at an angle of to the horizontal. The other rope has a tension of at an angle of to the horizontal.

The object is in equilibrium. What is the mass of the object?

53.

A uniform beam of weight is pivoted at P as shown. Weights of and are attached to its ends. The length of the beam is marked off at intervals. At which point should a further weight of be attached to achieve equilibrium?