Gravitation | ICSE Class 9 Physics Notes

This note covers gravitational attraction, the universal law of gravitation, its importance and applications, gravity, free fall, acceleration due to gravity, motion under gravity, mass and weight, gravitational units of force, and weight measurement.

What are gravitation and gravity?

Definition: Gravitation is the mutual attraction between objects because of their masses. Mass is the amount of matter in an object. A force is a push or pull arising from an interaction between objects.

Gravity is the gravitational attraction exerted by the Earth on objects. A dropped object moves towards the Earth because the Earth pulls it. Gravitation is the wider idea: the attraction is not restricted to the Earth and nearby objects.

Does gravitational attraction require contact?

A non-contact force acts without the interacting objects touching. Gravitation is a non-contact force. The Earth can attract an object above its surface even though the object is not touching the ground.

Gravitational force is always attractive: the interacting bodies pull towards one another. The attraction draws the interacting bodies towards one another without requiring physical contact between them.

A ball thrown vertically upwards slows down, stops momentarily at its highest position, and then falls. The Earth's attraction continues to act while the ball rises and while it falls. Upward motion does not mean that gravity acts upwards.

What changes when gravity acts?

Speed describes how quickly an object moves. Velocity describes its speed together with its direction. A force can change speed, direction, or both. Thus, a force can affect a moving object even when it does not bring that object to rest.

A falling fruit and the motion of the Moon around the Earth can both be understood through gravitational attraction. The same kind of interaction connects familiar motion near the ground with motion far from the Earth's surface.

What does the universal law of gravitation state?

Newton's universal law of gravitation states that every body attracts every other body with a force proportional to the product of their masses and inversely proportional to the square of their separation. The force acts along the line joining them.

Let F be the magnitude, or size, of the gravitational force; m₁ and m₂ the two masses; and r their separation. Let G be the universal gravitational constant, the proportionality constant that connects these quantities.

F = Gm₁m₂/r²

What conditions accompany the equation?

The equation applies directly to point masses, objects treated as having their mass concentrated at a point. For spherical bodies with a spherically symmetric mass distribution, use the distance between their centres when the bodies lie outside one another.

A spherically symmetric distribution has the same distribution of mass in every direction from its centre. The centre-to-centre separation is different from the gap between the surfaces. Substituting the surface gap would give the wrong force.

Derivation: How are the proportionalities combined?

1. Keeping the separation constant, the force is directly proportional to the product m₁m₂. Direct proportionality means that the quantities change in the same ratio.
2. Keeping both masses constant, the force is inversely proportional to r². This means that the force multiplied by the square of the separation remains constant.
3. Combining the two relations makes the force proportional to m₁m₂/r². Introducing the constant G converts this proportionality into an equation.

F = Gm₁m₂/r²

The inverse-square relation concerns the square of separation, not separation alone. State what remains constant when explaining either proportionality. Without that condition, a change in mass could occur alongside a change in separation.

What does the gravitational constant mean, and which units are used?

The International System of Units, abbreviated SI, provides the units used in these equations. The SI unit of mass is the kilogram, symbol kg. The SI unit of length is the metre, symbol m.

The SI unit of time is the second, symbol s. The SI unit of force is the newton, symbol N. A unit symbol following a number specifies the unit; the same letter used in an equation may represent a physical quantity.

Acceleration is the rate of change of velocity. The SI unit of acceleration is metre per second squared, written m/s². An acceleration of one metre per second squared corresponds to a velocity change of one metre per second in each second.

How is a newton defined?

One newton is the force that produces an acceleration of 1 m/s² in an object of mass 1 kg. For a constant mass, Newton's second law relates net force, mass and acceleration. The net force is the combined effect of all forces on the object.

Writing a for acceleration and m for the object's mass gives F = ma. Here F denotes the net force. Rearranging gives a = F/m, and the unit relation is 1 N = 1 kg m/s².

How is G different from an acceleration?

Rearranging the gravitational equation gives G = Fr²/(m₁m₂). Its SI unit is N m²/kg². The value used here is G = 6.67 × 10⁻¹¹ N m²/kg². This constant belongs to the universal law; it is not an object's acceleration.

Write units beside numerical answers. A force in newtons, a mass in kilograms and an acceleration in metres per second squared represent different quantities even when their numerical values happen to resemble one another.

Why is the universal law important?

The importance of the universal law lies in its common explanation of attraction near the Earth and attraction between astronomical bodies. A force of the same nature acts between the Earth and a falling object, between the Earth and Moon, and between the Sun and planets.

An orbit is the path followed by a body as it revolves around another. Gravitational attraction helps explain the Moon's motion around the Earth and the planets' motion around the Sun. The explanation does not require a separate kind of attraction for each pair.

Does the falling fruit attract the Earth?

Yes. Newton's third law states that when one object exerts a force on another, the second simultaneously exerts an equal and opposite force on the first. These two forces act on different objects.

The Earth pulls the fruit towards itself, and the fruit pulls the Earth towards itself. The two forces have equal magnitudes. They do not cancel when considering the fruit's motion, because the force exerted on the Earth does not act on the fruit.

The illustration shows a fruit beside a tree. A downward arrow is labelled “Force on the fruit by the Earth”, and an upward arrow is labelled “Force on the Earth by the fruit”.

See Fig. 6.33 in your NCERT textbook

Why is the Earth's motion not noticeable?

From a = F/m, equal forces do not generally produce equal accelerations in unequal masses. The Earth's mass is so large compared with the fruit's mass that its acceleration is extremely small. Its effect on the Earth is too small to be noticed.

This is different from saying that the fruit exerts no force or that the Earth's acceleration is exactly zero. The force comparison and the acceleration comparison answer different questions.

What are free fall and acceleration due to gravity?

Definition: Free fall is motion under gravitational force alone. Air resistance is the force exerted by air that opposes an object's motion through it. In the ideal free-fall description, air resistance is neglected.

The acceleration due to gravity is the acceleration caused by the Earth's gravitational force. Its magnitude is represented by the lowercase symbol g. Near the Earth's surface, use g = 9.8 m/s² unless another value is supplied.

The value of g can be taken to be nearly constant near the surface of the Earth. For quick estimations, g = 10 m/s² may be used. This approximation should be stated, and the same chosen value should be used throughout a calculation.

Does a greater mass mean a greater free-fall acceleration?

The acceleration due to the Earth's gravity does not depend on the mass of the falling object. A larger mass experiences a larger gravitational force, but acceleration depends on force divided by mass. Under the same free-fall conditions, this gives the same acceleration.

Distinguish the ideal condition from motion where other forces matter. If air resistance is significant, gravitational force alone is not the net force. The free-fall model then does not describe the complete force situation.

Successive positions appear beside a vertical metre scale. The labels give downward velocities of 0, 9.8, 19.6, 29.4 and 39.2 m/s at times 0, 1, 2, 3 and 4 s respectively.

See Fig. 4.10 in your NCERT textbook

The symbol m/s means metres per second, the SI unit of velocity. The diagram's velocity increases by equal amounts in successive equal time intervals. Its positions become farther apart, showing that constant acceleration does not mean constant speed.

Note: Uppercase G is the universal gravitational constant, with unit N m²/kg². Lowercase g is acceleration due to gravity, with unit m/s². They cannot be substituted for one another.

How are the equations of motion used for vertical free fall?

For motion in a straight line with constant acceleration, let u be initial velocity, v final velocity, t elapsed time, and s displacement. Displacement is the change in position, including its direction, from the starting point.

The letter s in these equations means displacement; the symbol s after a number means seconds. Velocity is measured in m/s, displacement in m and time in s. Keeping the quantity and its unit distinct helps prevent mistakes.

Which direction should be positive?

A sign convention is a choice of positive and negative directions. For a downward fall, choosing downwards as positive makes the acceleration +g. With upwards positive, the same downward acceleration is −g. Keep the choice unchanged throughout the solution.

With downwards positive and acceleration constant, the equations become v = u + gt, s = ut + ½gt², and v² = u² + 2gs. These equations relate different combinations of the same motion quantities.

Derivation: How is the velocity equation obtained?

1. For constant acceleration, acceleration equals change in velocity divided by elapsed time.
2. With downwards positive, write g = (v − u)/t for the downward acceleration.
3. Multiply by t to obtain gt = v − u, then add u to both sides.

v = u + gt

Derivation: How is the displacement equation obtained?

1. In a velocity-time graph, time is on the horizontal axis and velocity on the vertical axis. Constant acceleration gives a straight line.
2. Displacement equals the area between the graph and the time axis. For this downward motion, split it into a rectangle and a triangle.
3. The rectangle has area ut, and the triangle has area ½t(v − u). Replace v − u by gt and add the areas.

s = ut + ½gt²

What changes when the object is dropped?

Dropped from rest means u = 0 m/s. The equations then simplify to v = gt and s = ½gt². Distance travelled is the length of the path; in a straight downward fall, it equals the magnitude of displacement.

For an object thrown upwards, its velocity becomes zero momentarily at the highest point. Gravity still acts downwards. Zero velocity at that instant therefore does not imply zero acceleration or zero weight, the gravitational force on the object.

How can falling-object data be used in worked calculations?

A calculation should begin with the given data, the direction convention and the quantity required. Choose an equation that contains the required quantity and known values. Keep units in the substitution, and attach a direction to velocity or displacement where relevant.

How can acceleration be obtained from velocities?

Average acceleration measures the velocity change over a specified time interval divided by the interval's duration. It agrees with the acceleration at each instant when the acceleration is constant. Both the velocity difference and time difference must refer to the same interval.

Worked example 1. A falling object's downward velocity is 19.6 m/s at 2 s and 29.4 m/s at 3 s. Find its average acceleration during that interval, taking downwards as positive.

Formula: a = change in velocity / elapsed time.

Substitute: a = (29.4 − 19.6)/(3 − 2) = 9.8/1.

Answer: The average acceleration is 9.8 m/s² downwards. The calculation uses the difference between the two times, not the final time alone.

How are final velocity and distance found together?

Worked example 2. An object is dropped from rest and falls freely for 4 s before reaching the ground. Find its velocity and distance fallen after 4 s. Neglect air resistance and take g = 9.8 m/s² throughout, with downwards positive.

Formula: v = u + gt; s = ut + ½gt².

Substitute: u = 0 m/s; v = 0 + 9.8 × 4; s = 0 × 4 + ½ × 9.8 × 4².

Answer: The velocity is 39.2 m/s downwards, and the distance fallen is 78.4 m.

Velocity increases in proportion to elapsed time when an object falls from rest with constant g. Distance depends on the square of elapsed time. Do not multiply the final velocity by the whole falling time: that velocity was not maintained throughout the fall.

What happens over the first two seconds of the same fall?

Worked example 3. An object starts from rest and falls freely for 2 s. Find its velocity and distance fallen at that time. Neglect air resistance and use constant g = 9.8 m/s², taking downwards as positive.

Formula: v = gt; s = ½gt².

Substitute: v = 9.8 × 2; s = ½ × 9.8 × 2².

Answer: The velocity is 19.6 m/s downwards, and the distance fallen is 19.6 m. The equal numerical values have different units and represent different quantities.

How do mass and weight differ?

Weight is the gravitational force with which the Earth, or another attracting body, pulls an object. Mass describes the amount of matter in the object. Since weight is a force, its SI unit is newton; mass is measured in kilograms.

How is weight related to mass?

Let W denote the magnitude of weight. For an object of mass m in a location where acceleration due to gravity is g, W = mg. This follows from F = ma with gravitational acceleration substituted for a.

At a place where g remains constant, weight is directly proportional to mass. To find mass from a known weight, rearrange the same equation: m = W/g. A newton reading becomes a mass only after dividing by the appropriate gravitational acceleration.

Gravitational force can vary very slightly from place to place on the Earth, so weight can change while mass does not. For practical weighing, an object's weight remains almost the same everywhere on the Earth.

How is a small object's weight calculated?

A gram, symbol g after a number, is one-thousandth of a kilogram. This unit symbol is different from the quantity symbol g for acceleration due to gravity. Convert grams to kilograms before using W = mg with SI units.

Worked example 4. Find the weight of a 100 g mass where acceleration due to gravity is 9.8 m/s². Use 1 kg = 1000 g.

Formula: W = mg.

Substitute: m = 100/1000 = 0.1 kg; W = 0.1 × 9.8.

Answer: The weight is 0.98 N, directed downwards. Using g = 10 m/s² for a quick estimate would give about 1 N.

What are gravitational units of force?

A gravitational unit of force is defined through the weight of a specified mass under standard gravity. The kilogram-force and gram-force are force units. Their names refer to masses used in defining them, but they do not measure mass.

What do kilogram-force and gram-force mean?

A kilogram-force, written kgf, is the weight of a one-kilogram mass under standard gravity. A gram-force, written gf, is the weight of a one-gram mass under standard gravity. Standard gravity is the agreed reference acceleration used for these definitions.

For calculations using standard gravity rounded to 9.8 m/s², 1 kgf ≈ 9.8 N and 1 gf ≈ 0.0098 N. Also, 1 kgf = 1000 gf. The rounded newton conversions are approximate, not exact definitions.

Worked example 5. Express one kilogram-force in newtons, using standard gravity rounded to 9.8 m/s² and the definition that kilogram-force is the weight of a 1 kg mass under standard gravity.

Formula: W = mg.

Substitute: W = 1 × 9.8.

Answer: 1 kgf ≈ 9.8 N with the stated rounding. The 1 kg is the defining mass, while kgf and N both express force.

Do not write kg when the answer is a force. Likewise, do not write kgf for a mass. Before converting, identify whether the question asks for matter in an object or the gravitational pull on it.

How is weight measured with a spring balance?

A spring balance measures force through the stretching of a spring. One end of the spring is fixed; the other has a hook. Hanging an object from the hook stretches the spring, and a marked scale allows the force to be read.

The range is the interval of readings the instrument can measure. Its least count is the value represented by its smallest scale division. Check both before taking a reading, because different balances may have different ranges and divisions.

An illustration shows a vertical spring balance with a scale and a hook at its lower end. A stone hangs below the hook by a loop.

See Fig. 6.2 in your NCERT textbook

How should the scale be read?

1. Identify the scale marked in newtons when measuring weight as a force.
2. Check the maximum measurable weight and the value of each small division.
3. Suspend the object from the hook, ensuring that its weight does not exceed the instrument's range.
4. Read the scale carefully and record the weight with its unit.

Worked example 6. A spring balance has five equal divisions between the marks 0 N and 1 N. What force corresponds to one small division?

Formula: least count = difference between the marked values / number of equal divisions.

Substitute: least count = (1 − 0)/5.

Answer: Each small division represents 0.2 N. This is the least count of this scale, not a value to assume for every spring balance.

Why might the balance also show grams?

A spring balance usually also has a scale showing corresponding mass values in grams. These markings assume that the balance is being used on the Earth. The spring responds to force, while the additional scale converts that force into a mass reading.

A beam balance can determine mass by comparing an object's weight with that of a known mass. Distinguish the physical quantity being measured from a casual everyday use of the word “weight” for a value in kilograms.

Glossary

• Gravitation — The mutual attractive interaction between objects because they have mass.
• Gravity — The gravitational force with which the Earth attracts objects towards itself.
• Mass — The amount of matter in an object, measured in kilograms in SI.
• Weight — The gravitational force acting on an object, expressed in newtons in SI.
• Non-contact force — A force acting between objects without requiring the objects to touch.
• Gravitational constant — The constant G in the universal law, with SI unit N m²/kg².
• Acceleration — The rate at which an object's velocity changes with time.
• Acceleration due to gravity — The acceleration produced by gravitational attraction, represented by the lowercase symbol g.
• Free fall — Motion in which gravitational force alone acts on the moving object.
• Inverse-square relation — A relation in which gravitational force varies inversely with squared separation, keeping masses constant.
• Displacement — Change in an object's position from its starting point, including direction.
• Kilogram-force — The force equal to the weight of one kilogram under standard gravity.
• Spring balance — An instrument that measures force using the stretching of a spring.
• Least count — The value represented by the smallest division on an instrument's scale.

Common errors and misconceptions

• Misconception: Gravity requires contact with the ground. Correct: It is a non-contact force and acts on objects above the Earth's surface.
• Misconception: The universal law uses the gap between spherical surfaces. Correct: For the spherical bodies described here, use their centre-to-centre separation.
• Misconception: G and g are interchangeable. Correct: G is a gravitational constant; g is an acceleration. Their meanings and units differ.
• Misconception: A heavier freely falling body has greater acceleration. Correct: Gravitational acceleration is independent of the falling body's mass when other forces are neglected.
• Misconception: The Earth exerts more force on the fruit than the fruit exerts on the Earth. Correct: Their forces are equal; their accelerations differ because their masses differ.
• Misconception: Weight is measured in kilograms. Correct: Kilogram measures mass. Newton and kilogram-force measure force.
• Misconception: A ball's acceleration becomes zero at its highest point. Correct: Its velocity is momentarily zero, but gravity still accelerates it downwards.
• Misconception: The acceleration due to gravity is exactly constant everywhere. Correct: It can be treated as nearly constant near the Earth's surface for these calculations.

Exam-style questions with model answers

Q1. Define gravity and explain why it is a non-contact force. [2 marks]

1. Gravity is the gravitational force with which the Earth attracts objects towards itself.
2. It is a non-contact force because the Earth attracts objects even when they are not touching its surface.

Q2. State the universal law of gravitation, write its equation, define its symbols and explain the separation used for spherical bodies with a spherically symmetric mass distribution outside one another. [4 marks]

1. Every body attracts every other body with a force directly proportional to the product of their masses and inversely proportional to the square of their separation.
2. The force magnitude is F = Gm₁m₂/r², where F is force and m₁ and m₂ are the two masses.
3. G is the universal gravitational constant, and r is the separation between the point masses used in the equation.
4. For the specified spherical bodies, r is their centre-to-centre distance. The force acts along the line joining their centres.

Q3. A fruit falls towards the Earth, whose mass is much greater than the fruit’s mass. Explain the force exerted by each body, why these forces do not cancel on the fruit, and why the Earth's resulting motion is not noticeable. [4 marks]

1. The Earth attracts the fruit downwards, while the fruit attracts the Earth towards itself.
2. By Newton's third law, these gravitational forces have equal magnitudes and opposite directions.
3. They act on different objects. The force on the Earth does not cancel the force acting on the fruit.
4. Acceleration equals force divided by mass. The Earth's mass is so large compared with the fruit's mass that its acceleration is extremely small and its effect is too small to be noticed.

Q4. An object is dropped from rest and falls freely for 4 s before reaching the ground. Neglect air resistance and take constant g = 9.8 m/s². Choose downwards as positive, state the initial velocity and acceleration, calculate the final velocity and distance fallen, and explain why the final velocity cannot be used as the velocity throughout the fall. [5 marks]

1. With downwards positive, the initial velocity u is 0 m/s because the object is dropped from rest. Its acceleration is +9.8 m/s².
2. Use v = u + gt, where v is final velocity and t is elapsed time. Substitution gives v = 0 + 9.8 × 4 = 39.2 m/s downwards.
3. Use s = ut + ½gt² for displacement s. The zero initial velocity makes the first term zero, leaving s = ½ × 9.8 × 4².
4. This gives s = 78.4 m downwards. Since the object moves only downwards, its distance fallen is also 78.4 m.
5. The final velocity is reached only at the end. Velocity increases during the fall, so using the final velocity for the whole interval would overestimate the distance.

Q5. Distinguish mass from weight by meaning and SI unit. Then find the weight of a 100 g object where g = 9.8 m/s², using 1 kg = 1000 g. [3 marks]

1. Mass is the amount of matter in an object. Its SI unit is kilogram, and the given mass converts to 100/1000 = 0.1 kg.
2. Weight is the gravitational force acting on the object. Its SI unit is newton, and its magnitude is W = mg.
3. Substituting the stated values gives W = 0.1 × 9.8 = 0.98 N, acting downwards towards the Earth.

Q6. A kilogram-force is the weight of a 1 kg mass under standard gravity. Using standard gravity rounded to 9.8 m/s², convert 1 kgf to newtons and explain why kgf is not a mass unit. [2 marks]

1. Using W = mg gives W = 1 × 9.8 = 9.8 N, so 1 kgf is approximately 9.8 N with this rounding.
2. Kilogram-force measures gravitational force; kilogram measures mass. The defining mass does not make kilogram-force a mass unit.

Q7. A spring balance has five equal small divisions between 0 N and 1 N. Calculate its least count. Explain what the spring measures and the assumption behind an additional mass scale marked in grams. [3 marks]

1. The difference between the marked readings is 1 N. Dividing by five equal divisions gives a least count of 0.2 N per small division.
2. The spring measures force through its extension. When an object hangs from the balance, the force reading gives its weight.
3. The mass scale assumes use on the Earth. It shows the mass corresponding to a measured weight under that assumed gravitational attraction.

Key takeaways

• Gravitation is mutual attraction between masses; gravity is the Earth's gravitational pull on objects.
• The universal law connects force with the product of masses and the inverse square of their separation.
• Uppercase G is the universal gravitational constant, while lowercase g is acceleration due to gravity.
• Free fall involves gravitational force alone; neglect air resistance when applying the ideal model.
• Near the Earth's surface, gravitational acceleration can be treated as nearly constant for simple motion calculations.
• Mass measures matter in kilograms; weight measures gravitational force in newtons and is calculated using W = mg.
• Equal gravitational forces on two interacting bodies need not produce equal accelerations because their masses may differ.
• Use a consistent positive direction, define each symbol, convert units and state the direction of calculated motion quantities.

Test yourself

Why is gravity called a non-contact force?

The Earth can attract an object without touching it; contact with the ground is unnecessary.

In the inverse-square relation, what must remain constant?

The masses of both interacting bodies must remain constant when comparing force with their separation.

What distance belongs in the gravitational equation for the spherical bodies described here?

Use their centre-to-centre distance, rather than the gap between their outer surfaces.

Does zero velocity at the highest point of a vertical throw mean zero acceleration?

No. Velocity is momentarily zero, but gravitational acceleration continues to act downwards.

Why is the Earth's acceleration towards a falling fruit extremely small?

The Earth's very large mass gives it an extremely small acceleration under the mutual gravitational force.

What does “dropped from rest” tell you about initial velocity?

The initial velocity is zero, so the terms containing initial velocity vanish from the falling-body equations.

How do kg and kgf differ?

Kilogram is a unit of mass, whereas kilogram-force is a gravitational unit of force.

Why should you check each spring balance's smallest division?

Different instruments may have different least counts, so a familiar scale value cannot simply be assumed.