A Heavy Bob and a Light One Keep Exactly the Same Time
Learn the terms that describe a simple pendulum and their units, calculate time period and frequency from a count of oscillations, apply the formula relating period to length and gravity, and see what the period does and does not depend on.
Why does the weight of the bob make no difference to the timing?
Hang a heavy brass bob on a thread and time twenty swings. Replace it with a light wooden bob of the same size on a thread of the same length, and time twenty swings again.
The two times come out the same.
That is genuinely surprising the first time you see it. A heavier bob is pulled harder by gravity, so it ought to swing faster. But a heavier bob is also harder to get moving, and the two effects cancel exactly — the extra pull is matched by the extra reluctance to accelerate.
What the timing does depend on is the length. Shorten the thread and the swings quicken noticeably; lengthen it and they slow. The relationship turns out to be
with no mass in it anywhere.
The practical consequence is that a pendulum makes a reliable clock. It does not matter what the bob is made of or how hard you push it to start — only how long the thread is, and that does not change. A pendulum about a metre long takes about two seconds for a complete swing, and that steadiness is what a pendulum clock counts.
This page covers the third part of the ICSE Class 9 Physics chapter on measurements — the terms describing a pendulum, calculating period and frequency, using the formula, and what the period depends on.
The two times come out the same.
That is genuinely surprising the first time you see it. A heavier bob is pulled harder by gravity, so it ought to swing faster. But a heavier bob is also harder to get moving, and the two effects cancel exactly — the extra pull is matched by the extra reluctance to accelerate.
What the timing does depend on is the length. Shorten the thread and the swings quicken noticeably; lengthen it and they slow. The relationship turns out to be
with no mass in it anywhere.
The practical consequence is that a pendulum makes a reliable clock. It does not matter what the bob is made of or how hard you push it to start — only how long the thread is, and that does not change. A pendulum about a metre long takes about two seconds for a complete swing, and that steadiness is what a pendulum clock counts.
This page covers the third part of the ICSE Class 9 Physics chapter on measurements — the terms describing a pendulum, calculating period and frequency, using the formula, and what the period depends on.
What do oscillation, amplitude, time period and frequency mean?
Each names a different feature of the swing, and each has its own SI unit.
- Oscillation — one complete to-and-fro movement: from one extreme, through the mean position, to the other extreme, and back. It is a count, so it has no unit
- Amplitude — the maximum displacement from the mean position, measured to one side only. SI unit: metre ()
- Time period () — the time taken for one complete oscillation. SI unit: second ()
- Frequency () — the number of oscillations completed in one second. SI unit: hertz (), which is
- Effective length () — the distance from the point of suspension to the centre of gravity of the bob. SI unit: metre ()
Period and frequency are reciprocals:
Worked example. A pendulum with s has Hz. One with s has Hz. One with Hz has s.
The effective length is not the length of the thread. It runs to the centre of the bob, so
Worked example. A thread of cm carries a spherical bob of radius cm. The effective length is
This is the single commonest error in the whole chapter, and it matters because the formula in the third section uses and not the thread length. Using cm where cm belongs gives a period wrong in the second decimal place — small, but larger than the experiment's own uncertainty.
One oscillation is a full round trip, not a half. Going from the left extreme to the right extreme is half an oscillation. So a pendulum that passes the mean position times in a minute has completed oscillations, not — it crosses the middle twice per oscillation, once each way. Counting crossings instead of complete swings doubles the frequency, and questions are worded to test exactly that distinction.
- Oscillation — one complete to-and-fro movement: from one extreme, through the mean position, to the other extreme, and back. It is a count, so it has no unit
- Amplitude — the maximum displacement from the mean position, measured to one side only. SI unit: metre ()
- Time period () — the time taken for one complete oscillation. SI unit: second ()
- Frequency () — the number of oscillations completed in one second. SI unit: hertz (), which is
- Effective length () — the distance from the point of suspension to the centre of gravity of the bob. SI unit: metre ()
Period and frequency are reciprocals:
Worked example. A pendulum with s has Hz. One with s has Hz. One with Hz has s.
The effective length is not the length of the thread. It runs to the centre of the bob, so
Worked example. A thread of cm carries a spherical bob of radius cm. The effective length is
This is the single commonest error in the whole chapter, and it matters because the formula in the third section uses and not the thread length. Using cm where cm belongs gives a period wrong in the second decimal place — small, but larger than the experiment's own uncertainty.
One oscillation is a full round trip, not a half. Going from the left extreme to the right extreme is half an oscillation. So a pendulum that passes the mean position times in a minute has completed oscillations, not — it crosses the middle twice per oscillation, once each way. Counting crossings instead of complete swings doubles the frequency, and questions are worded to test exactly that distinction.
Formula
How do you calculate the time period from a count of oscillations?
Divide the total time by the number of oscillations:
Worked example 1. A pendulum completes oscillations in s:
Check the reciprocal: . Consistent.
Worked example 2. A pendulum makes oscillations in s:
Worked example 3. A pendulum makes oscillations in s:
Worked example 4 — working the other way. A pendulum of period s is left swinging for minutes. How many oscillations?
Worked example 5 — counting crossings. A bob passes its mean position times in s. Since each oscillation involves two crossings, that is oscillations:
**Reading the count as oscillations would halve the period to s — the trap from the previous section, in numbers.
Why you time many oscillations rather than one.** Starting and stopping a stopwatch carries a reaction-time error of perhaps s, and it lands on the total time however many swings were counted. Timing one oscillation of about s gives an error of roughly per cent. Timing oscillations gives a total of about s with the same s error — an error of per cent, and after dividing by the error in is only s.
So the fixed error is spread over every oscillation counted, and counting more is the cheapest way to improve the measurement. That is why a lab sheet always says "time 20 oscillations" rather than trusting a single swing — and it is the same reasoning behind averaging repeated readings in the previous part of this chapter.
Worked example 1. A pendulum completes oscillations in s:
Check the reciprocal: . Consistent.
Worked example 2. A pendulum makes oscillations in s:
Worked example 3. A pendulum makes oscillations in s:
Worked example 4 — working the other way. A pendulum of period s is left swinging for minutes. How many oscillations?
Worked example 5 — counting crossings. A bob passes its mean position times in s. Since each oscillation involves two crossings, that is oscillations:
**Reading the count as oscillations would halve the period to s — the trap from the previous section, in numbers.
Why you time many oscillations rather than one.** Starting and stopping a stopwatch carries a reaction-time error of perhaps s, and it lands on the total time however many swings were counted. Timing one oscillation of about s gives an error of roughly per cent. Timing oscillations gives a total of about s with the same s error — an error of per cent, and after dividing by the error in is only s.
So the fixed error is spread over every oscillation counted, and counting more is the cheapest way to improve the measurement. That is why a lab sheet always says "time 20 oscillations" rather than trusting a single swing — and it is the same reasoning behind averaging repeated readings in the previous part of this chapter.
How do you use the formula relating period, length and gravity?
**Substitute into , or rearrange it for whichever quantity is unknown:**
Take and .
Worked example 1 — finding the period. A pendulum has effective length m:
So a one-metre pendulum takes about two seconds per complete swing, and its frequency is about Hz.
Worked example 2 — a quarter of the length. For m:
Quartering the length halved the period — from s to s — because the length sits under a square root. This is the most useful single fact about the formula.
Worked example 3 — the other direction. For m:
Four times the length of worked example 1, and exactly twice the period. To double the period, quadruple the length.
Worked example 4 — the seconds pendulum. A seconds pendulum is defined as one whose time period is exactly s, so that each single swing takes one second. Its length is
That is why a pendulum clock stands about a metre tall.
**Worked example 5 — finding from an experiment.** A pendulum of effective length m is timed at oscillations in s, so s:
Close to the accepted , with the gap coming from timing and length uncertainties. **This is the standard school method for measuring , and it works because the pendulum converts a hard-to-measure acceleration into an easy-to-measure time.
Worked example 6 — finding the length for a target period.** What length gives s?
Agreeing with worked example 2, where m gave s.
The square root is where the arithmetic goes wrong. Doubling the length does not double the period; it multiplies it by . Check every answer against the one-metre benchmark: a pendulum shorter than a metre must have s, and a longer one must have s. That single comparison catches a misplaced square root immediately.
Take and .
Worked example 1 — finding the period. A pendulum has effective length m:
So a one-metre pendulum takes about two seconds per complete swing, and its frequency is about Hz.
Worked example 2 — a quarter of the length. For m:
Quartering the length halved the period — from s to s — because the length sits under a square root. This is the most useful single fact about the formula.
Worked example 3 — the other direction. For m:
Four times the length of worked example 1, and exactly twice the period. To double the period, quadruple the length.
Worked example 4 — the seconds pendulum. A seconds pendulum is defined as one whose time period is exactly s, so that each single swing takes one second. Its length is
That is why a pendulum clock stands about a metre tall.
**Worked example 5 — finding from an experiment.** A pendulum of effective length m is timed at oscillations in s, so s:
Close to the accepted , with the gap coming from timing and length uncertainties. **This is the standard school method for measuring , and it works because the pendulum converts a hard-to-measure acceleration into an easy-to-measure time.
Worked example 6 — finding the length for a target period.** What length gives s?
Agreeing with worked example 2, where m gave s.
The square root is where the arithmetic goes wrong. Doubling the length does not double the period; it multiplies it by . Check every answer against the one-metre benchmark: a pendulum shorter than a metre must have s, and a longer one must have s. That single comparison catches a misplaced square root immediately.
What does the time period depend on, and what does it ignore?
**It depends on the effective length and on , and on nothing else — not the mass, not the material, and not the amplitude for a small swing.
It depends on the length as .** Four times the length gives twice the period, as worked examples 1 and 3 showed — m gives s and m gives s.
**It depends on as .** A smaller gives a longer period, since sits in the denominator. On the Moon, where is roughly a sixth of its value on Earth, the same pendulum would take about times as long per swing — so a pendulum clock carried there would run slow, losing time steadily.
The same effect appears on Earth. is very slightly smaller at the equator and on a mountain than at sea level near a pole, so a pendulum clock adjusted in one place runs a little wrong in another.
It does NOT depend on the mass of the bob. A brass bob and a wooden bob of the same size on equal threads keep identical time, because the extra gravitational pull on the heavier one is exactly offset by its greater resistance to being accelerated. No mass appears in the formula, and that is not an approximation.
It does NOT depend on the amplitude, for a small swing. Pull the bob cm aside or cm aside and the period is the same. This is called isochronism, and it is what makes a pendulum usable as a clock: as friction slowly reduces the amplitude, the timing does not drift.
But the amplitude independence has a limit. It holds well for small swings — roughly under about ten degrees from the vertical — and for a large swing the period becomes slightly longer. So the correct statement is that the period is independent of amplitude provided the amplitude is small, and a question asking for the conditions under which the formula applies is asking for exactly that qualification.
Worked comparison. Two pendulums, one with a g bob and one with a g bob, both of effective length m. Both have
Identical, because the mass never entered the calculation.
Worked comparison of two lengths. Effective lengths m and m give s and s. **The ratio of periods is and the ratio of lengths is — the square-root relationship, visible in two numbers.
The independence of mass is the same physics as free fall. All objects fall with the same acceleration regardless of mass, for the identical reason: gravitational pull and resistance to acceleration both scale with mass, so their ratio does not. A pendulum is a controlled version of a falling object**, which is why it can be used to measure at all, and why the mass cancels out of both experiments.
It depends on the length as .** Four times the length gives twice the period, as worked examples 1 and 3 showed — m gives s and m gives s.
**It depends on as .** A smaller gives a longer period, since sits in the denominator. On the Moon, where is roughly a sixth of its value on Earth, the same pendulum would take about times as long per swing — so a pendulum clock carried there would run slow, losing time steadily.
The same effect appears on Earth. is very slightly smaller at the equator and on a mountain than at sea level near a pole, so a pendulum clock adjusted in one place runs a little wrong in another.
It does NOT depend on the mass of the bob. A brass bob and a wooden bob of the same size on equal threads keep identical time, because the extra gravitational pull on the heavier one is exactly offset by its greater resistance to being accelerated. No mass appears in the formula, and that is not an approximation.
It does NOT depend on the amplitude, for a small swing. Pull the bob cm aside or cm aside and the period is the same. This is called isochronism, and it is what makes a pendulum usable as a clock: as friction slowly reduces the amplitude, the timing does not drift.
But the amplitude independence has a limit. It holds well for small swings — roughly under about ten degrees from the vertical — and for a large swing the period becomes slightly longer. So the correct statement is that the period is independent of amplitude provided the amplitude is small, and a question asking for the conditions under which the formula applies is asking for exactly that qualification.
Worked comparison. Two pendulums, one with a g bob and one with a g bob, both of effective length m. Both have
Identical, because the mass never entered the calculation.
Worked comparison of two lengths. Effective lengths m and m give s and s. **The ratio of periods is and the ratio of lengths is — the square-root relationship, visible in two numbers.
The independence of mass is the same physics as free fall. All objects fall with the same acceleration regardless of mass, for the identical reason: gravitational pull and resistance to acceleration both scale with mass, so their ratio does not. A pendulum is a controlled version of a falling object**, which is why it can be used to measure at all, and why the mass cancels out of both experiments.
Exam tip
Exam tip: use the effective length, and time twenty swings
**Effective length thread length radius of the bob.** A cm thread with a cm radius bob gives cm — never cm.
Convert to metres before substituting: cm is m. Mixing centimetres into the formula with in gives an answer out by a factor of ten.
One oscillation is a full round trip. A bob crossing the mean position times has made 25 oscillations.
**** and — check the reciprocal as a free verification.
**Time oscillations, not one.** The stopwatch error lands on the total, so dividing by divides the error by .
Learn the benchmark: a m pendulum has s. Every answer should be checked against it — shorter means under s, longer means over.
**, so quadrupling** the length doubles the period. Doubling the length multiplies it by , not by .
**A seconds pendulum has s and cm — quote both.
Rearrange before substituting**: and , with .
State the independences explicitly when asked: independent of mass and of amplitude (for small amplitudes), dependent on length and **.
And say "provided the amplitude is small"** — the unqualified claim loses the mark.
Convert to metres before substituting: cm is m. Mixing centimetres into the formula with in gives an answer out by a factor of ten.
One oscillation is a full round trip. A bob crossing the mean position times has made 25 oscillations.
**** and — check the reciprocal as a free verification.
**Time oscillations, not one.** The stopwatch error lands on the total, so dividing by divides the error by .
Learn the benchmark: a m pendulum has s. Every answer should be checked against it — shorter means under s, longer means over.
**, so quadrupling** the length doubles the period. Doubling the length multiplies it by , not by .
**A seconds pendulum has s and cm — quote both.
Rearrange before substituting**: and , with .
State the independences explicitly when asked: independent of mass and of amplitude (for small amplitudes), dependent on length and **.
And say "provided the amplitude is small"** — the unqualified claim loses the mark.
Did you know
Why a pendulum clock would lose time on a mountain
A pendulum clock does not keep time by any internal mechanism deciding what a second is. It keeps time by counting swings, and the length of a swing is set by the length of the pendulum and by .
So anything that changes changes the clock.
Carry the clock up a mountain and falls slightly, because you are further from the centre of the Earth. Since , a smaller makes each swing take longer, so the clock counts fewer swings per real hour — it loses time.
Move it from a pole towards the equator and the same thing happens for two reasons at once: the Earth bulges at the equator so you are further from the centre, and the rotation reduces the effective there as well.
The effect is small on Earth, but it runs the other way in a big way off it. On the Moon, with about a sixth of Earth's, the same pendulum takes roughly times as long per swing — a clock reading two hours when six have passed.
The temperature matters too, through the other variable. A metal pendulum rod expands when warm, increasing , and makes the swing slower. So a pendulum clock runs slow in summer and fast in winter, which is why better clocks used rods of materials chosen to expand as little as possible, or compensating arrangements that hold the effective length steady.
What makes all of this interesting rather than merely inconvenient is that the sensitivity works both ways. A clock that is disturbed by a change in is also an instrument for measuring a change in — and the same formula rearranged is exactly how worked example 5 of this page found from a stopwatch and a metre rule.
So anything that changes changes the clock.
Carry the clock up a mountain and falls slightly, because you are further from the centre of the Earth. Since , a smaller makes each swing take longer, so the clock counts fewer swings per real hour — it loses time.
Move it from a pole towards the equator and the same thing happens for two reasons at once: the Earth bulges at the equator so you are further from the centre, and the rotation reduces the effective there as well.
The effect is small on Earth, but it runs the other way in a big way off it. On the Moon, with about a sixth of Earth's, the same pendulum takes roughly times as long per swing — a clock reading two hours when six have passed.
The temperature matters too, through the other variable. A metal pendulum rod expands when warm, increasing , and makes the swing slower. So a pendulum clock runs slow in summer and fast in winter, which is why better clocks used rods of materials chosen to expand as little as possible, or compensating arrangements that hold the effective length steady.
What makes all of this interesting rather than merely inconvenient is that the sensitivity works both ways. A clock that is disturbed by a change in is also an instrument for measuring a change in — and the same formula rearranged is exactly how worked example 5 of this page found from a stopwatch and a metre rule.
Exam relevance
How does the simple pendulum feed into JEE Main and NEET?
Because the pendulum is the standard first example of simple harmonic motion, and it is also the classic experiment for error analysis.
This is the foundation for Class 11 Physics Oscillations, examined in JEE Main and NEET. The formula is derived there rather than quoted, as a special case of
and the condition flagged on this page — that the amplitude must be small — becomes the explicit approximation used in that derivation. A student who knows the formula only holds for small swings understands why the approximation is needed, which is the point assertion-reason questions on this topic test.
The same chapter compares it with the spring. A mass on a spring has , which does depend on mass — so the pendulum's independence of mass is not a general feature of oscillators but a specific consequence of gravity supplying the restoring force. Questions pairing the two are a recurring JEE Main type, and the contrast is the point.
Error analysis uses this experiment more than any other. Since , the relative error in is the relative error in plus twice the relative error in — because appears squared. So halving the timing error is worth twice as much as halving the length error, and the time 20 oscillations rule from this page is the practical consequence. That calculation is asked directly in JEE Main, and it builds straight on the least-count work in the previous part of this chapter.
**Where itself reappears.** Class 11 Gravitation explains why varies with height, with depth and with latitude — the mountain and equator effects described above — and both JEE Main and NEET examine it. The pendulum is the measuring instrument that makes those variations detectable.
For NEET Physics, the pendulum appears as a numerical on period, frequency and length, and as a conceptual question on what the period depends on.
What the questions look like. For board work, expect define the terms with units, **find and from a count of oscillations, apply the formula in all three rearrangements, the seconds pendulum length, and state what the period depends on with reasons. For JEE Main and NEET**, expect numericals with the square-root relationship, comparisons with a spring, and percentage-error calculations on .
How board and competitive emphasis differ. A board paper rewards the definitions with units and the substitution written out. A competitive paper assumes the formula and tests the proportional reasoning — recognising in one step that quadrupling doubles , without computing either.
The single trap that costs the most marks. Using the thread length as the effective length. The effective length runs to the centre of gravity of the bob, so a cm thread with a cm radius bob gives cm. The error is under two per cent and therefore looks like a plausible answer — which is exactly why questions specify the bob's radius. If a problem tells you the radius of the bob, it is telling you to add it.
This is the foundation for Class 11 Physics Oscillations, examined in JEE Main and NEET. The formula is derived there rather than quoted, as a special case of
and the condition flagged on this page — that the amplitude must be small — becomes the explicit approximation used in that derivation. A student who knows the formula only holds for small swings understands why the approximation is needed, which is the point assertion-reason questions on this topic test.
The same chapter compares it with the spring. A mass on a spring has , which does depend on mass — so the pendulum's independence of mass is not a general feature of oscillators but a specific consequence of gravity supplying the restoring force. Questions pairing the two are a recurring JEE Main type, and the contrast is the point.
Error analysis uses this experiment more than any other. Since , the relative error in is the relative error in plus twice the relative error in — because appears squared. So halving the timing error is worth twice as much as halving the length error, and the time 20 oscillations rule from this page is the practical consequence. That calculation is asked directly in JEE Main, and it builds straight on the least-count work in the previous part of this chapter.
**Where itself reappears.** Class 11 Gravitation explains why varies with height, with depth and with latitude — the mountain and equator effects described above — and both JEE Main and NEET examine it. The pendulum is the measuring instrument that makes those variations detectable.
For NEET Physics, the pendulum appears as a numerical on period, frequency and length, and as a conceptual question on what the period depends on.
What the questions look like. For board work, expect define the terms with units, **find and from a count of oscillations, apply the formula in all three rearrangements, the seconds pendulum length, and state what the period depends on with reasons. For JEE Main and NEET**, expect numericals with the square-root relationship, comparisons with a spring, and percentage-error calculations on .
How board and competitive emphasis differ. A board paper rewards the definitions with units and the substitution written out. A competitive paper assumes the formula and tests the proportional reasoning — recognising in one step that quadrupling doubles , without computing either.
The single trap that costs the most marks. Using the thread length as the effective length. The effective length runs to the centre of gravity of the bob, so a cm thread with a cm radius bob gives cm. The error is under two per cent and therefore looks like a plausible answer — which is exactly why questions specify the bob's radius. If a problem tells you the radius of the bob, it is telling you to add it.
Key takeaways
The simple pendulum, period and frequency: quick revision
- Oscillation: one complete to-and-fro swing — no unit, it is a count.
- Amplitude: maximum displacement from the mean position, in metres.
- **Time period : time for one oscillation, in seconds. Frequency : oscillations per second, in hertz** ().
- **** and : s gives Hz; Hz gives s.
- **Effective length thread length radius of bob, to the bob's centre of gravity**, in metres. A cm thread with a cm bob gives cm.
- One oscillation is a full round trip — crossings of the mean position is 25 oscillations.
- ****: in s gives s, Hz; in s gives s and Hz; in s gives s and Hz.
- A pendulum of s makes oscillations in three minutes.
- **Time oscillations, not one** — the stopwatch error lands on the total, so dividing by divides the error by .
- ****, with and . Take , .
- ** m gives s** — the benchmark to check every answer against.
- m gives s; m gives s. ****, so quadrupling doubles .
- **A seconds pendulum has s**, so cm.
- **Measuring **: m with oscillations in s gives s and .
- For s, m.
- Depends on: effective length () and ().
- Does NOT depend on: the mass or material of the bob, or the amplitude — provided the amplitude is small.
- Mass cancels because the extra gravitational pull on a heavier bob is offset by its greater resistance to acceleration — the same reason all objects fall alike.
- Isochronism — amplitude independence — is what makes a pendulum a usable clock as friction reduces the swing.
- For a large amplitude the period is slightly longer, so always say "for small amplitudes".
- **A clock runs slow where is smaller** — on a mountain, near the equator, and far more so on the Moon, by about .
- A warm metal rod lengthens, so the clock runs slow in summer.
Tie a stone to a thread about a metre long, time twenty swings, and work out — then see how close a thread and a stopwatch bring you to .
- Amplitude: maximum displacement from the mean position, in metres.
- **Time period : time for one oscillation, in seconds. Frequency : oscillations per second, in hertz** ().
- **** and : s gives Hz; Hz gives s.
- **Effective length thread length radius of bob, to the bob's centre of gravity**, in metres. A cm thread with a cm bob gives cm.
- One oscillation is a full round trip — crossings of the mean position is 25 oscillations.
- ****: in s gives s, Hz; in s gives s and Hz; in s gives s and Hz.
- A pendulum of s makes oscillations in three minutes.
- **Time oscillations, not one** — the stopwatch error lands on the total, so dividing by divides the error by .
- ****, with and . Take , .
- ** m gives s** — the benchmark to check every answer against.
- m gives s; m gives s. ****, so quadrupling doubles .
- **A seconds pendulum has s**, so cm.
- **Measuring **: m with oscillations in s gives s and .
- For s, m.
- Depends on: effective length () and ().
- Does NOT depend on: the mass or material of the bob, or the amplitude — provided the amplitude is small.
- Mass cancels because the extra gravitational pull on a heavier bob is offset by its greater resistance to acceleration — the same reason all objects fall alike.
- Isochronism — amplitude independence — is what makes a pendulum a usable clock as friction reduces the swing.
- For a large amplitude the period is slightly longer, so always say "for small amplitudes".
- **A clock runs slow where is smaller** — on a mountain, near the equator, and far more so on the Moon, by about .
- A warm metal rod lengthens, so the clock runs slow in summer.
Tie a stone to a thread about a metre long, time twenty swings, and work out — then see how close a thread and a stopwatch bring you to .