# Physics Tutorial: Types of Forces I. Gravitational Force and Weight

In this Physics tutorial, you will learn:

• What is gravitational force?
• What is gravitational field and how does it affect the value of gravitational force?
• How to calculate the gravitational force at different heights?
• What is weight and how does it differ from the gravitational force?
• How does weight and mass differ from each other?
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4.2Types of Forces I. Gravitational Force and Weight

## Introduction

Suppose you are resting on a balance. Does the balance show any value?

Now think about this scenario. You attach the balance below your feet and jump from a wall with the balance attached. Does the balance show any value while you are flying?

In the first case, the balance shows a certain value, while in the second, it does not show anything. We will explain the reasons for this in the following paragraphs.

In this tutorial, we will discuss extensively about two forces that are often confused with each other. They are gravitational force and weight. So, let's see how to deal with them.

## What is Gravitational Force? What is Gravitational Field?

In the previous Physics tutorial "What Causes the Motion? The Meaning of Force", we stated that there are two kind of forces considering the way how they act on objects: contact and field forces. The first category includes forces exerted by an object to another when there is a contact between them, while the later includes forces exerted when objects have a certain distance between them. Such forces are not exerted directly but by means of the respective field generated through them. For example, electric force is caused by the electric field generated by charged particles, magnetic force is caused by the magnetic field generated by magnets and so on.

Likewise, by definition, Gravitational Force is the attracting force between any two objects due to the interaction between their respective gravitational fields.

The phenomenon of objects' attraction due to the presence of the gravitational field (and therefore, of gravitational force) is known as "gravity". For example, Earth attracts all objects near its surface because these objects are inside the range of its gravitational field. Hence, we can say "Gravitational Field is the space around an object in which its gravitational attraction effect is observed." This means that Gravitational Force is a distant (field) force as mentioned above.

## What are the factors affecting the Gravitational Force?

From our experience, we know (as stated before) that Earth attracts us. Thus, when we jump, we fall again on the ground due to this attraction. However, it is a fact that we attract the Earth as well, although the effect caused by the Earth attraction in our body is more noticeable than the attraction effect caused by our body on the Earth, due to the difference in size between the Earth and us. Therefore, the effect of gravitational attraction is mutual.

The fact that the gravitational attraction of Earth is more noticeable than that of any other object near its surface, makes us think (rightfully) that the gravitational force depends on the mass of the objects involved. Thus, more massive the object, greater the gravitational field (and therefore, the resulting gravitational force) produced. This means gravitational force is directly proportional to the product of masses of the objects involved.

Another factor affecting the magnitude of Gravitational Force between two objects is the distance between them. It is obvious that the farther away the objects be, weaker the resulting gravitational force produced is. This means the Gravitational Force is inversely proportional to the product of the distance between the two objects.

Mathematically, we can write:

### Equation 1a

Fg = G × m1 × m2/R2

where m1 and m2 are the masses of the objects involved, and R is the distance between the objects (from centre to centre, not from surface to surface).

The quantity G is a constant. It is known as the gravitational constant. Its numerical value is

G = 6.674 × 10-11 N × m2/kg2

The value of constant G has been calculated experimentally.

Look at the figure below:

If one object is much heavier than the other, we write M and m for their masses instead of m1 and m2, where M stands for the heaviest object and m for the lightest one.

### Example 1

What is the gravitational force exerted by the Earth at a 1 kg object resting on its surface? Take the mass of the Earth equal to 5.972 × 1024 kg and the Radius of the Earth equal to 6371 km.

### Solution 1

Clues:

M = 5.972 × 1024 kg
m = 1 kg
R = 6371 km = 6 371 000 m = 6.371 × 106 m
G = 6.674 × 10-11 N × m2/kg2

Thus, since

### Equation 1b

Fg = G × M × m/R2

we obtain after substituting the known values,

Fg = 6.674 × 10-11 × 5.972 × 1024 × 1/(6.371 × 106)2N
= 39.86 × 1013/40.59 × 1012
= 0.981 × 101 N
= 9.81N

## What Quantity Characterizes Mathematically the Gravitational Field?

In the previous example, we saw that a 1 kg object is attracted by 9.81 N gravitational force. From the equation (1.b) it is obvious that if the mass of the object was 2 kg, the gravitational force produced by the Earth would be 2 × 9.81N = 19.62 N, for a 3 kg object the gravitational force is 3 × 9.81 N = 29. 43 N and so on. Therefore, the relationship between the object's mass m and the resulting gravitational force Fg produced by the Earth near its surface is

### Equation 2

Fg = m × 9.81

The factor 9.81 is not new. In the Kinematics chapter we stated that the value of gravitational acceleration near the Earth surface is g = 9.81 m/s2. If using the dimensional analysis discussed in the Physics tutorial "Length, Mass and Time. Dimensional Analysis", we can find that the coefficient 9.81 obtained in the equation (2) has the following dimensions:

9.81 = Fg/m
=>the unit of the constant 9.81 is [N/kg]
= [kg × m/s2 × kg]
= [m/s2]

(By definition, 1 N is equal to 1kg × m/s2)

Therefore, we obtained the same unit as that of acceleration. Hence, we can conclude that the constant 9.81 we found earlier, is nothing more but the gravitational acceleration (the acceleration of free fall discussed in Kinematics). Thus, the equation (2) for the gravitational force becomes

### Equation 3

Fg = m × g

In Dynamics, i.e. in the part of Physics dealing with forces, the gravitational acceleration g is otherwise known as the "gravitational field strength." Therefore, depending on the context, g can be expressed in different names.

To summarize, gravitational field strength (gravitational acceleration) g is the physical quantity, which characterizes mathematically the gravitational field. Its value is g = 9.81 m/s2 = 9.81 N/kg near the Earth surface but it becomes smaller when moving away from the Earth. It can be calculated by the equation

### Equation 4

g = Fg/m

where m is the mass of the object, (not the Earth). Substituting the equation (1.b) in the numerator instead of Fg, we obtain

### Equation 5

g = G × M × m/R2/m = >
g = G × M/R2

From the equation (5) it is obvious that the gravitational field strength g generated by the Earth, does not depend on the properties of the other object which is inside this gravitational field, but only on the distance R from the centre of the Earth (the other quantities such as mass of the Earth M and the gravitational constant G are constants).

### Example 2

What is the magnitude of gravitational field strength g at 3629 km above the Earth surface? Take Mass of the Earth equal to 5.972 × 1024 kg and Radius of the Earth equal to 6371 km.

### Solution 2

In this example, we have to calculate the height h from the ground as shown in the figure.

We must add h to the Earth's Radius R in the equation (5), to express the distance between the objects. Thus, giving that h = 3629km = 3 629 000m = 3.629 × 106m, we have

### Equation 6

g = G × M/(R + h)2
= 6.674 × 10-11 × 5.972×1024/(6.371×106 + 3.629 × 106 )2
= 6.674 × 10-11 × 5.972 × 1024/(10 × 106)2
= 6.674 × 10-11 × 5.972 × 1024/( 107 )2
= 39.857 × 1013/1014
= 39.857 × 10-1
= 3.9857 m/s2
≈ 4m/s2

This result means that at h = 3629 km above the Earth's surface, the attraction of gravity is only 40% of its value on the ground. This is the main reason why there is no air at that height, i.e. the Earth is not able to hold the air at such heights and as a result, the air escapes in the space. (More than 99% of the air is in the first 50 km of the atmosphere).

## What is Weight?

By definition, Weight, W is the pushing force exerted by an object on the place it lies (or the pulling force when hanged on a string), caused by the gravity.

From the above definition, it is obvious that weight depends on the gravity, so many people often confuse Weight and Gravitational Force with each other. Despite the common things (and the same result obtained in most cases when calculating these two forces), there are some very important differences between them. They are:

1. Weight is a contact force while gravitational force is a field one. This means when an object is flying in the air, it has no weight, as it is not exerting any pushing force on the ground (remember the example in the Introduction). However, there is a mutual gravitational force exerted by the object and the Earth, which causes the object to fall on the ground if no lifting force is available.
2. Even when the object is on the ground, sometimes weight and gravitational force may not be numerically equal. This occurs for example when the object lies on a slope as shown in the figure.

As you can see, Weight is always normal to the surface in which the object lies while gravitational force is always vertical (downward). If the slope is inclined at an angle θ to the horizontal direction, we have for the magnitude of weight.

### Equation 7

|W| = |Fg| × cos θ = |m × g| × cos θ

(From geometry, it is known that two angles with perpendicular sides are equal. Therefore, the angle formed by the slope and the horizontal direction is equal to the angle formed by the weight vector and the vertical direction because horizontal direction is perpendicular to the vertical one and weight vector is perpendicular to the slope).

Remark! Unlike the gravitational force vector, the weight vector does not begin at the centre of the object because its application point is at the place of contact between the two objects, i.e. at the bottom face of the object (and as a result, at the upper face of the slope).

### Example 3

A 20 kg object rests on a slope as shown in the figure.

#### Calculate:

1. Gravitational force exerted by the Earth on the object
2. Weight of the object

Take g = 9.81 m/s2, cos 200 = 0.940 and sin 200 = 0.342

### Solution 3

First, we draw the vectors of weight and gravitational force acting on the object.

Gravitational force is:

Fg = m × g
= 20 kg × 9.81 m/s2
= 196.2 N

Weight of the object (in magnitude) is:

|W| = |Fg| × cos θ
= |m × g| × cos θ
= 196.2 N × 0.940
= 184.4 N

As you see, the results are different.

## The Difference between Weight and Mass

People very often confuse weight and mass. This is because when an object rests on a balance, it shows the mass in kilogram. However, this is a trick used by the balance constructors to avoid problems since people are more familiar to the concept of mass. Even the popular terminology used in this case is confusing. We say "This object weighs 50 kg" instead of "The mass of this object is 50 kg."

In fact, the balance measures the weight in Newtons but when stamping units in the balance, the constructors convert them in kilograms by dividing weight with the gravitational field strength. Therefore, a balance shows 60 kg instead of 60 × 9.81 = 588.6 N when a 60 kg object rests on it.

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