30
2.1Forces and their direction
A force, which we commonly associate with a pull or a push, is a vector quantity. We represent forces with arrows whose length is proportional to the magnitude of the force. It is important that we are able to correctly identify the direction of forces.
Forces
In this chapter we will deal with the following forces: weight, string tension, spring tension, normal contact forces, drag forces, buoyant forces (upthrust) and frictional forces.
Weight
Weight is the gravitational force between a body and the planet the body is on. The weight of a body is given by the formula
W=mg
where m is the mass of the body and g is the gravitational field strength of the planet. The unit of g is newton per kilogram, N kgâ1. The gravitational field strength is also known as âthe acceleration due to gravityâ or the âacceleration of free fallâ. Therefore the unit of g is also m sâ2.
If m is in kg and g in N kgâ1 or m sâ2 then W is in newtons, N. On the surface of the earth, g = 9.81 N kgâ1âa number that we will sometime approximate by the more convenient 10 N kgâ1. (We almost always do this for multiple choice exam questions.) This force is always directed vertically downward, as shown in Figure 2.1.

Figure 2.1: The weight of an object is always directed vertically downward.

The mass of an object is the same everywhere in the universe, but its weight depends on the location of the body. For example, a mass of 70 kg has a weight of 687 N on the surface of the earth (g = 9.81 N kgâ1) and a weight of 635 N at a height of 250 km from the earthâs surface (where g = 9.07 N kgâ1). However, on the surface of Venus, where the gravitational field strength is only 8.9 N kgâ1, the weight is 623 N.
Tension
The force that arises in any body when it is stretched or compressed is called tension. A string that is taut is said to be under tension. This force is the result of interactions between the molecules of the material making up the string and is electrical in nature. A tension force in a string is created when two forces are applied in opposite directions at the ends of the string (see Figure 2.2).

Figure 2.2: A tension force in a string.

To say that there is tension in a string means that an arbitrary point P on the string is acted upon by two forces (the tension T) as shown in Figure 2.3. If the string hangs from a ceiling and a mass m is tied at the other end, tension develops in the string. At the point of support at the ceiling, the tension force pulls down on the ceiling and at the point where the mass is tied the tension acts upwards on the mass.
31

Figure 2.3: The tension is directed along the string.

In most cases we will idealize the string by assuming it is massless. This does not mean that the string really is massless, but rather that its mass is so small compared with any other masses in the problem that we can neglect it. In that case, the tension T is the same at all points on the string. The direction of the tension force is along the string. Further examples of tension forces in a string are given in Figure 2.4. A string or rope that is not taut has zero tension in it.

Figure 2.4: More examples of tension forces.

Springs
A spring that is pulled so that its length increases will develop a tension force inside the spring that will tend to bring the length back to its original value. Similarly, if it is compressed a tension force will again try to restore the length of the spring, Figure 2.5. Experiments show that for a range of extensions of the spring, the tension force is proportional to the extension, T = kx, where k is known as the spring constant. This relation between tension and extension is known as Hookeâs law.

Figure 2.5: Tension is proportional and opposite to the extension or compression.

CHECK YOURSELF 1
A block is attached to a vertical spring. Will the extension of the spring be different on the earth and on the moon?
Normal contact forces
If a body touches another body, there is a normal contact force between the two bodies. This force is perpendicular to the surface of the body exerting the force, hence the name normal. Like tension, the origin of this force is also electrical. In Figure 2.6 we show the normal force R on several bodies.
32

Figure 2.6: Examples of normal forces, R.

We can understand the existence of contact forces in a simple model in which atoms are connected by springs. The block pushes down on the atoms of the table, compressing the springs under the block. This creates the normal force on the block, Figure 2.7.

Figure 2.7: A simple model of contact forces.

Drag forces
Drag forces are forces that oppose the motion of a body through a fluid (a gas or a liquid). Typical examples are the air resistance force experienced by a car (see Figure 2.8) or plane, or the resistance force experienced by a steel marble dropped into a jar of honey. Drag forces are directed opposite to the velocity of the body and in general depend on the speed and shape of the body. The higher the speed, the higher the drag force.

Figure 2.8: Drag forces oppose motion.

For the case of a spherical body of radius r falling through a fluid of density Ï with speed v the drag force is given by Stokeâs law:
FD=6ÏÎ·rv
where Î· is a quantity known as the viscosity of the fluid. (This formula applies to spherical, small objects moving through a fluid smoothly.) Viscosity is a measure of the forces between layers of the fluid that are moving relative to each other; water has a viscosity of about 10â3 Pa s, motor oil 0.5 Pa s and honey 1.5 Pa s. The unit is Pa s where the pascal (Pa) is the unit of pressure and equals N mâ2. Viscosity decreases with increasing temperature.
Buoyant forces (upthrust)
Any object placed in a fluid (liquid or gas) experiences an upward force called the buoyant force (see Figure 2.9). This force is given by
FB=ÏgVimm
where Vimm is the volume of the body that is immersed in the fluid.
If the buoyant force equals the weight of the body, the body will float in the fluid. If the buoyant force is less than the weight, the body will sink. The buoyant force is caused by the pressure that the fluid exerts on the body. The pressure is greater at the bottom compared to that at the top.

Figure 2.9: Buoyant force.

Frictional forces
Frictional forces generally oppose the motion of a body. They are also electrical in origin. (See Figure 2.10.)
33

Figure 2.10: Examples of frictional forces, f. In a there is motion to the right, which is opposed by a single frictional force that will eventually stop the body. In b the force accelerating the body is opposed by a frictional force. In c the body does not move, but it does have a tendency to move down the plane, and so a frictional force directed up the plane opposes this tendency, keeping the body in equilibrium.

Friction arises whenever one body slides over another. In this case we have dynamic friction. Friction also arises whenever there is a tendency for motion, not necessarily motion itself. For example, a block that rests on an inclined plane has a tendency to slide down the plane, so there is a force of friction up the plane. Similarly, if you pull on a block on a level rough road with a small force the block will not move. This is because a force of friction develops that is equal and opposite to the pulling force. In this case we have static friction.
In the simple model of matter consisting of atoms connected by springs, pushing the block to the right results in springs stretching and compressing. The net result is a force opposing the motion, friction, Figure 2.11.

Figure 2.11: Friction in the simple atoms-and-springs model of matter.

A more realistic model involves irregularities (called asperities) in the surfaces which interlock, opposing sliding, Figure 2.12.

Figure 2.12: Exaggerated view of how asperities oppose the sliding of one surface over the other.

Frictional forces are still not very well understood, and there is no theory of friction that follows directly from the fundamental laws of physics. However, a number of simple, empirical âlawsâ of friction have been discovered. These are not always applicable and are only approximately true, but they are useful in describing frictional forces in general terms.
The âfriction lawsâ may be summarised as follows:
1the area of contact between the two surfaces does not affect the frictional force.
2the force of dynamic friction is equal to fd = Î¼d R, where R is the normal force between the surfaces and Î¼d is the coefficient of dynamic friction.
3the force of dynamic friction does not depend on the speed of sliding.
4the maximum force of static friction that can develop between two surfaces is given by fs = Î¼s R, where R is the normal force between the surfaces and Î¼s is the coefficient of static friction with Î¼s > Î¼d.
EXAM TIP
One of the most common mistakes is to think that Î¼sR is the formula that gives the static friction force. This is not correct. This formula gives the maximum possible static friction force that can develop between two surfaces.
Figure 2.13 shows how the frictional force f varies with a pulling force F. The force F pulls on a body on a horizontal rough surface. Initially the static frictional force matches the pulling force and we have no motion, fs = F. When the pulling force exceeds the maximum possible static friction force, Î¼s R, the frictional force drops abruptly to the dynamic value of Î¼d R and stays at that constant value as the body accelerates. This is a well-known phenomenon of everyday life: it takes a lot of force to get a heavy piece of furniture to start moving (you must exceed the maximum value of the static friction force), but once you get it moving, pushing it along becomes easier (you are now opposed by the smaller dynamic friction force).
34

Figure 2.13: The variation of the frictional force f between surfaces with the pulling force F.

CHECK YOURSELF 2
A block of weight 5.0 N rests on a rough horizontal table. The static coefficient of friction between the block and the table is 0.60. A horizontal force of 2.2 N is applied to the block. What is the frictional force on the block?
Other forces
In later chapters we will deal with other forces, such as the electric force, the magnetic force, and the nuclear force.
Free-body force diagrams
A free-body force diagram is a diagram showing the magnitude and direction of all the forces acting on a chosen body. The body is shown on its own, free of its surroundings and of any other bodies it may be in contact with. In Figure 2.14 we show three situations in which forces are acting; below each is the corresponding free-body force diagram for the shaded bodies.

Figure 2.14: Free-body diagrams.

In any mechanics problem, it is important to be able to draw correctly the free-body force diagrams for all the bodies of interest. It is also important that the length of the arrow representing a given force is proportional to the magnitude of the force.
WORKED EXAMPLE 2.1
aDraw the free-body diagram for a pendulum when the bob is at the extreme left position.
bDraw the free-body diagram for a ball that moves on the inside surface of a vertical circle at the three positions shown in Figure 2.15.

Figure 2.15: For worked example 2.1.

Answer
aSee Figure 2.16.

Figure 2.16: The two forces on the bob.

35
bSee Figure 2.17. The size of the normal force here will be understood when we move to Newtonâs second law. For now make sure you understand the direction.

Figure 2.17: The forces on the ball.

Equilibrium
Equilibrium of a point particle means that the resultant or net force on the particle is zero. The net force on a particle is the one single force whose effect is the same as the combined effect of individual forces acting on the particle. We denote it by âF. Finding the net force is easy when the forces are in the same or opposite directions. In Figure 2.18a, the net force is (we take the direction to the right to be positive)
âF=12+6.0â8.0=10Â N.
This is positive, indicating a direction to the right. In Figure 2.18b it is (we take the direction upward to be positive)
âF=5.0+6.0â4.0â8.0=â1.0Â N
The negative sign indicates a direction vertically down.

Figure 2.18: The net force is found by plain addition and/or subtraction when the forces are in the same or opposite direction.

WORKED EXAMPLE 2.2
Determine the magnitude of the force F given that the block (Figure 2.19) is in equilibrium.

Figure 2.19: For worked example 2.2.

Answer
For equilibrium, âF = 0 and so 6.0 + F + 6.0 â 15 = 0. This gives F = 3.0 N.
WORKED EXAMPLE 2.3
A brick of weight 50 N rests on a horizontal surface. The coefficient of static friction between the brick and the surface is 0.60, and the coefficient of dynamic friction is 0.20. A horizontal force F is applied to the brick, its magnitude increasing uniformly from zero. Once the brick starts moving the pulling force no longer increases. Estimate the net force on the moving brick.
Answer
The maximum frictional force that can develop between the brick and the surface is fs = Î¼s R, which evaluates to 0.60 Ã 50 = 30 N. So motion takes place when the pulling force is just barely larger than 30 N. Once motion starts the frictional force will be equal to Î¼d R, i.e., 0.20 Ã 50 = 10 N. The net force on the brick in that case will be just larger than 30 â 10 = 20 N.
36
CHECK YOURSELF 3
An ice cube floats in water in a glass. What happens to the level of the water when the ice cube melts?
WORKED EXAMPLE 2.4
An iceberg floats in seawater. The density of ice is 920 kg mâ3 and that of seawater is 1020 kg mâ3. Estimate the fraction of the volume of the iceberg that is under water.
Answer
The weight of the iceberg is balanced by the buoyant force. The weight is given by
W=mg=ÏicegV
and the buoyant force is
FB=ÏwaterÂ gVimm.
The weight and the buoyant forces are equal and so
ÏiceÂ gV=ÏwaterÂ gVimmÂ 
VimmÂ V=ÏiceÂ ÏwaterÂ =9201020=0.90
WORKED EXAMPLE 2.5
A small sphere of radius r and density Ïbody falls vertically through a viscous fluid of viscosity Î· and density Ïfluidwith constant speed v. (As we will be discussing extensively later on, motion with constant velocity is equivalent to equilibrium, and the net force is zero.)
Show that the speed v is given by v=29(ÏbodyÂ âÏfluid)r2gÎ·.
Answer
The forces on the falling sphere are the weight vertically downward and the drag force and the buoyant force upward.
Hence W = B + FD. Now W = mg = Ïbody gV, B = Ïfluid gV and FD = 6ÏÎ·rv so that:
ÏbodyÂ gV=ÏfluidÂ gV+6ÏÎ·rv.
Now use the formula for the volume of a sphere, V=4Ïr33, to get
ÏbodyÂ g4Ïr336ÏÎ·rvv=ÏfluidÂ g4Ïr33+6ÏÎ·rv=4Ïr3g3(ÏbodyÂ âÏfluidÂ )=29(ÏbodyÂ âÏfluidÂ )r2gÎ·
CHECK YOURSELF 4
A car moves on a straight horizontal road with constant speed. The force due to the engine pushing the car forward is F. Is the total resistive force on the car less than F, equal to F, or greater than F?
TEST YOUR UNDERSTANDING
1Figure 2.20 shows a block at rest on a rough table. The block is connected by a string that goes over a pulley to a second hanging block. Draw the forces on each body.

Figure 2.20: For question 1.

37
2aA bead is at rest at the top of a sphere, Figure 2.21. Draw the forces on the bead.
bThe bead is given a small push, and at the right-most position shown, it is just about to lose contact with the sphere. Draw the forces on the bead.
cDraw the forces on the bead in the intermediate position.

Figure 2.21: For question 2.

3Look at Figure 2.22. Compare the tension in the string in the two cases.

Figure 2.22: For question 3.

4A spring is compressed by a certain distance and a mass is attached to its right end, as shown in Figure 2.23. The mass is at rest on a rough table. On a copy of the diagram, draw the forces acting on the mass.

Figure 2.23: For question 4.

5A mass hangs attached to three strings, as shown in Figure 2.24. On a copy of the diagram, draw the forces on:
athe hanging mass
bthe point where the strings join.

Figure 2.24: For question 5.

6A force of 12 N extends a spring by 3 cm. What force will extend the spring by 4 cm?
7Two identical springs, each with a spring constant of k = 220 N mâ1, are connected to a trolley and fixed supports as shown in Figure 2.25.
aWhen the trolley is in equilibrium the springs have their natural length. Calculate the net force on the trolley when it is moved 2.0 cm to the right.
bWhen the trolley is in equilibrium each spring is extended by 4.0 cm. Calculate the net force on the trolley when it is moved 2.0 cm to the right.

Figure 2.25: For question 7.

8Find the net force on each of the bodies, AâF, shown in Figure 2.26. The only forces acting are the ones shown. Indicate direction by ârightâ, âleftâ, âupâ and âdownâ.
38

Figure 2.26: For question 8.

9A small ball falls through a liquid at constant speed. Copy Figure 2.27, then draw and label the forces acting on the ball.

Figure 2.27: For question 9.

10Explain why it is impossible for a mass to hang attached to two horizontal strings as shown in Figure 2.28.

Figure 2.28: For question 10.

11A block X rests on top of another block Y (see Figure 2.29). Both blocks are on a frictionless, horizontal table. There is friction between the two blocks. In a, a horizontal force is applied to block Y. In b, the force is applied to block X. In both cases, the two blocks move together without sliding on each other.
In each case draw the horizontal forces acting on each block.

Figure 2.29: In a the force is applied to Y. In b it is applied to X. The blocks move together.

12A mass of 2.00 kg rests on a rough horizontal table. The coefficient of static friction between the block and the table is 0.60. The block is attached to a hanging mass by a string that goes over a smooth pulley, as shown in Figure 2.30. Determine the largest mass that can hang in this way without forcing the block to slide.

Figure 2.30: For question 12.

13A boy tries to lift a suitcase of weight 220 N by pulling upward on it with a force of 140 N. The suitcase does not move. Calculate the normal force from the floor on the suitcase.
14A block of weight 15 N rests on a horizontal table. A man pushes the block downward with a force of 12 N. What is the normal force of the table on the block?
