In this explainer, we will learn how to apply Newton’s second law of motion to define gravitational field strength as the force acting on an object per kilogram of its mass.

The mass of an object can refer to the constant of proportionality between the force that acts on an object and the acceleration of the object. This can be expressed as and as where is the force applied, is the mass of the object, and is the acceleration of the object.

This is how mass is defined in Newton’s second law of motion.

The mass of an object is also related to the gravitational force produced by the
object. This mass is termed *gravitational mass*.

It is not obvious that objects in general produce gravitational forces, but it is very obvious that Earth produces gravitational forces, as unsupported objects noticeably accelerate toward Earth.

The reason that the gravitational forces produced by Earth are clearly observed, but the gravitational forces produced by other objects are not clearly observed is that the mass of Earth is very great compared to objects such as humans, or even much more massive objects such as large buildings.

The mass of Earth is approximately kg. Producing a gravitational force sufficiently great to accelerate objects at a rate clearly observable by human senses requires a mass comparable to the enormous mass of Earth.

The mass of Earth produces what is termed a *gravitational field*. A gravitational
field is a region within which an object is acted on by a gravitational
force.

The mass of any object produces a gravitational field, but for objects on Earth, the forces due to their gravitational fields are sufficiently small that they are approximately zero. When considering objects on Earth, for most examples, it is valid to model Earth as the only object to produce a gravitational field.

The gravitational force on an object in the gravitational field of Earth is called the weight of the object. The weight of an object is related to its mass by the following formula: where is the weight of the object, is the mass of the object, and is the gravitational field strength at the position of the object.

Weight is a force, so its SI unit is newton.

The SI base unit of mass is the kilogram.

In nonscientific English, the weight of an object is commonly referred to as being measured in kilograms. The correct scientific use of English, however, requires that a number of kilograms refer to a measurement of the mass of an object. The weight of an object is measured in newtons rather than in kilograms.

The formula can be rearranged to make the subject, giving

We see from this that the SI unit of gravitational field strength is newtons per kilogram (N/kg).

On Earth, the gravitational field strength is approximately 9.8 N/kg.

The weight of an object that is supported by a surface acts on that surface. The following figure shows an object with a weight, , at rest on a surface.

A force with a magnitude of is exerted on the surface.

Let us look at an example that involves the relationship between mass and weight.

### Example 1: Determining the Weight of an Object from its Mass

A chair has a mass of 50 kg. What force does the chair’s weight apply to the ground below it?

### Answer

The weight force on the object is given by the formula where the mass of the object, , is 50 kg and the gravitational field strength, , is 9.8 N/kg.

We have then

A force of 490 N is exerted on the ground below the chair.

The value of the gravitational field strength on Earth,
9.8 N/kg,
is the same as the value of the acceleration due to gravity on Earth,
9.8 m/s^{2}.

The weight of an object is a force on the object that produces an acceleration of
9.8 m/s^{2} toward
Earth.

It is important to notice that the gravitational force on objects due to the
gravitational field of Earth produces the same magnitude of acceleration on all
objects independent of the masses of these objects. On Earth, all objects are
accelerated by gravity at
9.8 m/s^{2},
whatever the masses of these objects are.

The gravitational field strength on other planets need not be 9.8 N/kg. For example, on Earth’s moon the gravitational field strength is approximately 1.6 N/kg.

Let us look at an example involving the weight force of an object on the Moon.

### Example 2: Determining the Weight of an Object on the Moon

An astronaut who has a mass of
81.25 kg goes to the
Moon, where the acceleration due to gravity is
1.6 m/s^{2}. What force does the astronaut’s weight apply to the lunar surface
beneath the astronaut’s feet?

### Answer

The weight force on the object is given by the formula where the mass of the object, , is 50 kg and the gravitational field strength, , is 1.6 N/kg.

We have then

Let us now look at an example in which the gravitational field strength on a planet is determined.

### Example 3: Determining Gravitational Field Strength Using the Mass and Weight of an Object

What is the gravitational field strength on a planet where an object with a mass of 25 kg has a weight of 300 N?

### Answer

The weight force on the object is related to the mass of the object and the gravitational field strength by the formula where the is the weight force, is the mass, and is the gravitational field strength.

To determine the gravitational field strength, the formula must be rearranged to make the subject. This can be done by dividing the formula by :

Substituting the values in the question, we have

An unsupported object would accelerate toward this planet at
12 m/s^{2}.

The gravitational field strength of Earth decreases as distance from Earth increases. A great increase in the distance from Earth is required to decrease the gravitational field strength sufficiently for the change in the weight force on an object to be clearly observable by human senses.

It is important to understand that the mass of an object is independent of the position of the object in any gravitational field.

We can imagine a spacecraft that is in deep space, sufficiently far away from any other objects, that the gravitational field strength at the position of the spacecraft is approximately zero.

The weight force on the spacecraft at this location is given by where the mass of the spacecraft is and where .

We can see that hence, zero weight force acts on the spacecraft.

We can now suppose that the spacecraft has an engine that can exert a force on the spacecraft to accelerate it.

The acceleration of the spacecraft due to the force provided by its engine is given by Newton’s second law of motion,

The force required from the spacecraft engine to accelerate the spacecraft by
1 m/s^{2}
is given by

This is shown in the following figure.

If the spacecraft was on Earth and was accelerating horizontally, then
the force required from the engine to accelerate the spacecraft horizontally by
1 m/s^{2}
would be given by

This is shown in the following figure.

The horizontal force on the spacecraft is the same force as required when the spacecraft is in deep space, where the gravitational field strength is approximately zero. The gravitational field strength does not change the relationship between the mass of the spacecraft and the force required from the engine to accelerate it.

On Earth, where the gravitational field strength is approximately 9.8 N/kg, the weight force on the spacecraft is given by

In order to accelerate vertically upward at
1 m/s^{2},
the spacecraft engine would have to produce a vertically upward force given by

This is shown in the following figure.

The additional force required from the engine is not due to a change in the mass of the spacecraft, it is because the weight force on the spacecraft acts in the opposite direction of the force from the engine. The mass of the spacecraft is the same on Earth as it is in deep space.

Let us look at an example involving the weight force on an object at a great distance from Earth.

### Example 4: Comparing the Change in the Mass and the Weight of an Object due to Change in Gravitational Field Strength

An astronaut on Earth, where the gravitational field strength is 9.8 N/kg, has a mass of 65 kg and a weight of 637 N. The astronaut is sent to a space station, where the gravitational field strength is 9.5 N/kg.

- What is the astronaut’s mass on the space station?
- What is the astronaut’s weight on the space station? Give your answer to the nearest newton.

### Answer

The mass of an object is independent of the position of the object in a gravitational field, so the mass of the astronaut on the space station is equal to the mass of the astronaut on Earth, which is 65 kg.

The weight force on the object is related to the mass of the object and the gravitational field strength by the formula where the is the weight force, is the mass, and is the gravitational field strength.

We have shown that the mass of the astronaut on the space station is 65 kg, and the gravitational field strength at the position of the space station is 9.5 N/kg.

We see than that

To the nearest newton, this is 618 N.

Let us now summarize what has been learned in this explainer.

### Key Points

- The weight force acting on an object depends on the mass of the object and the gravitational field strength at the position of the object. This relationship can be expressed by the formula where is the weight force that acts on the object, is the mass of the object, and is the gravitational field strength at the position of the object.
- The SI unit for weight is the newton, and the SI unit for mass is the kilogram.
- The SI unit for gravitational field strength is newtons per kilogram (N/kg).
- The gravitational field strength at the surface of Earth is 9.8 N/kg.
- The gravitational field strength at a point in newtons per kilogram is equal to the acceleration due to gravity at that point in metres per second squared.
- All objects produce gravitational fields.
- The gravitational field strength around objects that are not extremely massive (such as planets) is sufficiently small that it is approximately zero.
- The mass of an object is independent of its position in a gravitational field.