Contents

# Weight|Definition & Meaning

## Definition

Weight is defined as the gravitational force acting on the body in the downward direction. It measures how much gravity pulls the object according to its composed matter.

**Figure 1** shows the **weight** of an object which depends upon how much matter the object is made of and the **gravitational** pull on it.

## Gravity

**Weight** cannot be understood without discussing **gravity.** It is the main reason we can** stand** and move on the Earth.

The **attractive force** between two **bodies** is called gravity. A bigger body has a greater **force** of attraction than a smaller body.

Newton’s **law** of **gravitation,** expressed mathematically, is:

F = G × $\mathsf{\dfrac{{m_1}{m_2}}{r^2}}$

Where,

m_{1} = mass of the **first** object

m_{2} = mass of the **second** object

**r** = **distance** from the **centers** of first and second objects

**G** = **gravitational** constant = 6.67 × 10^{-11} Nm^{2}.kg^{-2}

**Figure** **2** shows the parameters of the law of gravitation.

## Parameters in Weight’s Mathematical Formula

### Force

Force is the **external** influence on a body to change its **position.** It changes the **velocity** of the object, causing the body to **accelerate.**

The **direction** of force is in the direction of **acceleration** produced, hence is a **vector** quantity.

If a body of **mass** **m** is applied with a **force F**, it produces **acceleration** **a** in the body. Mathematically, it can be expressed as

**F = ma**

This is also known as Newton’s **second law** of motion. **Figure** **3** shows its illustration.

The SI **unit** of force is **Newtons (N)** which is equal to kilograms meters per second squared (kg.ms^{-2}).

### Mass

The mass is defined as the **matter** confined in an object. It is denoted by **m,** and its SI unit is **kilograms (kg).**

### Gravitational Acceleration

Gravitational acceleration is the **change** in **velocity** caused by force due to **gravity.**

**Planets** have different **gravitational fields,** so the value of **g** is different for each. **Table 1** shows the value of **g** for the nine planets.

Name of Planets | g (m/s^{2}) |

Mercury | 3.7 |

Venus | 8.87 |

Earth | 9.8 |

Mars | 3.69 |

Jupiter | 23.12 |

Saturn | 8.96 |

Neptune | 11.0 |

Uranus | 8.69 |

Pluto | 0.66 |

**Table 1**

## Mathematical Formula for Weight

According to Newton’s** second law** of **motion**:

F = ma

If the object **falls freely** on the ground, then

F = F_{g} , a = g

So the **equation** becomes

F_{g} = mg

As the **gravitational** force F_{g} is equal to the **weight W** of an object, the above equation can be written as:

**W = mg**

The **weight** depends upon two factors; the **mass m** and the gravitational acceleration** g** experienced by the object. It is demonstrated in **figure 4**.

The unit of **weight** is the same as the force that is **Newtons.**

## Difference Between Weight and Mass

**Weight** and **mass** are often perceived as similar but are two **different** quantities. The **mass** does not change, whereas **weight** changes when **gravitational** acceleration **g** changes.

For example, a car’s **weight** on Jupiter will be greater than its weight on Earth as the value of **g** is greater for **Jupiter **(**23.12 ms**^{-2}) as compared to **Earth **(**9.8 ms**^{-2}). But its **mass** will remain the **same.**

**Weight** is a **vector** quantity having a direction the same as the direction of **g,** whereas **mass** is **scalar**.

## Weighing Scales

People often confuse weight with mass, as the scales which measure mass are called “**weighing scales**.” The term “**weight**” is more common among people than “**mass**.”

These scales measure the **force** exerted on them due to **gravity** which is usually the same everywhere as we live on the planet **Earth.**

The **units** in which weight or what should be called **mass** is measured are as follows:

### Metric Units

The **scales** that **weigh** mass are measured in kilograms (**kg**), grams(**g**), and tonne(**t**). They are related as follows:

1 kg = 1000 g

1 t = 1000 kg = 1,000,000 g

### US Units

The **US** has separate **units** measuring mass in pounds, ounces, and tons. They are related as follows:

1 pound = 16 ounces

1 ton = 2000 pounds = 32,000 ounces

## Apparent Weight

So far, we have discussed the **real weight W**. Apparent weight is defined as the **upward reaction** of the gravitational **force** acting on an object. It is also known as **tension** and is denoted by **T**.

**Apparent weight** changes with the object’s **motion,** whereas the real weight remains the same.

Consider a **person** standing in an **elevator** on a **weighing** scale. Four cases of the object’s **motion** are discussed:

### Case 1

Consider the person and the elevator at **rest** or the lift moving with **uniform velocity.** In this case, **acceleration** will be **0**. The net **force** acting on the body will be

F_{net} = T – W

ma = T – W

As a = 0, so

0 = T – W

**T = W**

In this case, the **apparent** weight will be **equal** to the **real** weight.

### Case 2

If the elevator moves in an **upward** direction with **acceleration** **a**, the net **force** on the body will be

F_{net} = T – W

ma = T – W

**T = W + ma**

The **apparent** weight is now the **real** weight of the object plus the **upward force** acting on it. Hence the apparent weight is **more** than the real weight.

### Case 3

If the elevator goes **down** with acceleration **a**, the **apparent** weight of the person **decreases** by **ma** as the signs of **W** and **T** change.

F_{net} = W – T

ma = W – T

**T = W – ma**

### Case 4

If the elevator **free falls**, its acceleration equals the **gravitational** acceleration. The F_{net} will be the same as for the elevator going **downwards,** so

F_{net} = W – T

ma = mg – T

Substituting **a = g**,

mg = mg – T

Subtracting **mg** on both sides, so

**T = 0**

**Figure 5** shows the four cases of apparent weight.

## An Example of Calculating an Object’s Weight

An object of mass **20 kg** is placed on the **Earth** and the planet **Mars.** Calculate its **weight** on the planets Earth and Mars.

### Solution

The equation for **weight** is:

W = mg

Here, **m = 20 kg**. The value of **g is 9.8 ms ^{-2}** on planet Earth. Its

**weight**on the

**Earth**will be:

W = (20)(9.8)

**W = 196 N**

The value of **g** on Mars from **Table 1** is **3.69 ms ^{-2}**. Its

**weight**on

**Mars**will be:

W = (20)(3.69)

**W = 73.8 N**

*All the images are created using GeoGebra.*