Oscillations: Simple Harmonic Motion and Pendulum Notes
Physics notes on oscillations for exams: simple harmonic motion, velocity and energy, the laws of the simple pendulum, second's pendulum and resonance.
By GK24 Editorial Team· Published · 4 min read
A motion that repeats itself in equal intervals of time is periodic motion; when the body moves to and fro about a fixed point, the motion is also oscillatory. A swing, a pendulum, the balance wheel of a watch, a loaded spring and the atoms of a solid about their lattice sites all oscillate. The simplest and most important case is simple harmonic motion, in which the restoring force on the body is proportional to its displacement from the mean position and acts towards that position. Almost every question set from this chapter comes from three places: the two formulas for time period, the behaviour of a pendulum when something about it changes, and the places where kinetic energy, potential energy, velocity and acceleration reach their maximum and minimum values.
Simple harmonic motion and its quantities
In simple harmonic motion the restoring force is given by F equal to minus kx, where x is the displacement and k the force constant, so the acceleration is minus omega squared times x: it is always directed towards the mean position and is largest where the displacement is largest. The displacement varies with time as x equal to A sine of omega t plus the phase constant. The amplitude A is the greatest displacement on either side; the time period T is the time of one complete oscillation; the frequency f is the number of oscillations in one second, measured in hertz; and the angular frequency omega equals two pi f, which is also two pi divided by T. The velocity is omega times the square root of A squared minus x squared, so it is greatest at the mean position, where it equals A omega, and zero at the extreme positions. The acceleration behaves in the opposite way: zero at the mean position and greatest at the extremes, where it equals A omega squared. For a mass m on a spring of force constant k, the time period is two pi times the square root of m divided by k, which is the one case in which mass does affect the period.
The simple pendulum
A simple pendulum is a heavy point mass hung by a light inextensible string from a rigid support. For small oscillations its time period is two pi times the square root of the length divided by the acceleration due to gravity. Four laws follow, and examiners test them one at a time.
| Change made | Effect on time period |
|---|---|
| Mass of the bob increased | No change; the period is independent of mass |
| Amplitude slightly increased | No change for small oscillations; this is the law of isochronism |
| Length made four times | Period becomes twice, since T varies as the square root of length |
| Taken to the moon | Period increases about 2.45 times, as g is about one sixth |
| Taken from the equator to the poles | Period decreases, because g is greater at the poles |
| Taken to the centre of the earth or into a satellite | No oscillation at all, because g is zero there |
| Lift accelerating upwards | Period decreases, effective g being g plus a |
| Lift accelerating downwards or in free fall | Period increases; in free fall the pendulum stops oscillating |
A pendulum whose time period is exactly two seconds is called a second's pendulum, and its length is about 99.4 centimetres, that is roughly one metre, at a place where g is 9.8 metres per second squared. Because length rises with temperature, a pendulum clock goes slow in summer and fast in winter, and clockmakers use invar, an alloy with a very small coefficient of expansion, to reduce the error. Galileo first noticed that the period of a swinging lamp did not depend on how far it swung, and Christiaan Huygens used the principle to build the first pendulum clock in 1656.
Energy in simple harmonic motion
The energy of an oscillator keeps changing form while the total stays the same if there is no friction. Kinetic energy is greatest at the mean position and zero at the extremes; potential energy is greatest at the extremes and zero at the mean position; and the sum, the total energy, is one half of k times A squared, which is also one half of m omega squared A squared. Two points follow that examiners like: the total energy is proportional to the square of the amplitude, so doubling the amplitude makes the energy four times, and both the kinetic and the potential energy complete two cycles in the time the displacement completes one, so their frequency is twice the frequency of the oscillation.
Damped and forced oscillations, and resonance
Real oscillations die away because friction and air resistance take energy from the system; this is damped oscillation, and the amplitude falls steadily while the period is almost unchanged. If an outside periodic force is applied, the body finally oscillates with the frequency of that force, which is forced oscillation. When the frequency of the applied force equals the natural frequency of the body, the amplitude becomes very large: this is resonance. It explains why a column of soldiers is ordered to break step while crossing a bridge, why a radio set picks out one station when its circuit is tuned to the station's frequency, why a glass can be shattered by a singer's note and why a hollow box amplifies the sound of a sitar or a violin.
Exam Point of View
Three kinds of questions come up. First, the formulas: the period of a pendulum and of a spring, the length of a second's pendulum, and what happens when the length is made four times. Second, the change-and-effect questions: the pendulum in a lift, on the moon, at the poles, in free fall, or with a heavier bob, where the trap is to assume that mass matters. Third, the maximum and minimum positions of velocity, acceleration, kinetic energy and potential energy, often as a matching question. Resonance is asked through its examples, chiefly soldiers breaking step on a bridge and the tuning of a radio.
Important Facts
| Simple harmonic motion | F equals minus kx; acceleration equals minus omega squared times x |
|---|---|
| Period of a simple pendulum | T equals 2 pi root of l by g |
| Period of a loaded spring | T equals 2 pi root of m by k; independent of g |
| Second's pendulum | Time period 2 seconds; length about 99.4 cm where g is 9.8 |
| Velocity in SHM | omega root of A squared minus x squared; maximum A omega at the mean position |
| Acceleration in SHM | Maximum A omega squared at the extreme positions, zero at the mean position |
| Total energy | Half of k A squared; proportional to the square of the amplitude |
| Frequency of energy change | Kinetic and potential energy vary at twice the frequency of the oscillation |
| Pendulum on the moon | Period becomes about 2.45 times, as g is about one sixth |
| Pendulum in a lift | Period decreases going up, increases going down, no oscillation in free fall |
| Pendulum clock and season | Slow in summer as the rod lengthens; fast in winter; invar reduces the error |
| Law of isochronism | Discovered by Galileo; the period does not depend on amplitude for small swings |
| First pendulum clock | Built by Christiaan Huygens in 1656 |
| Unit of frequency | Hertz; one hertz is one oscillation per second |
| Resonance | Driving frequency equals natural frequency; amplitude becomes very large |
Practice MCQs on this topic
If a particle is moving with simple harmonic motion, the velocity is _________ at the mean position.
- A.Maximum
- B.Zero
- C.Minimum
- D.None of these
Show answer
In simple harmonic motion the velocity is given by omega times the square root of A squared minus x squared.
Read the full explanation →A child is sitting on a swing. Its minimum and maximum heights from the ground are 0.75 m and 2 m respectively. Its maximum speed will be
- A.10 m/s
- B.5 m/s
- C.8 m/s
- D.15 m/s
Show answer
A swing is a pendulum, and its speed is greatest at the lowest point, where all the potential energy gained at the top has turned into kinetic energy.
Read the full explanation →The time period of a simple pendulum does not depend on which of the following?
- A.The length of the pendulum
- B.The mass of the bob
- C.The acceleration due to gravity
- D.All of these
Show answer
The time period of a simple pendulum is two pi times the square root of the length divided by g, in which the mass of the bob does not appear at all; a heavy bob and a light bob of the same length keep the same time, which is the…
Read the full explanation →The length of a second's pendulum, at a place where the acceleration due to gravity is 9.8 m/s squared, is nearly
- A.1 metre
- B.2 metres
- C.0.5 metre
- D.9.8 metres
Show answer
A second's pendulum is one whose time period is exactly two seconds, so that each swing from one side to the other takes one second.
Read the full explanation →If the length of a simple pendulum is made four times its original value, its time period will become
- A.Four times
- B.Two times
- C.Half
- D.One fourth
Show answer
The time period varies as the square root of the length, so multiplying the length by four multiplies the period by the square root of four, that is by two; to double the period of a pendulum its length must be made four times.
Read the full explanation →If a pendulum clock is taken from the earth to the moon, where the acceleration due to gravity is about one sixth, its time period will
- A.Decrease
- B.Increase
- C.Remain the same
- D.Become zero
Show answer
The time period varies inversely as the square root of g, so if g falls to one sixth, the period is multiplied by the square root of six, about 2.45 times; the pendulum swings more slowly and a pendulum clock taken to the moon would lose time badly.
Read the full explanation →In simple harmonic motion, the potential energy of the oscillating body is maximum at
- A.The mean position
- B.The extreme positions
- C.A point midway between the mean and extreme positions
- D.Every point of the path
Show answer
Potential energy is proportional to the square of the displacement, so it is greatest where the displacement is greatest, that is at the two ends of the path, where the body is momentarily at rest and its kinetic energy is zero.
Read the full explanation →The acceleration of a body executing simple harmonic motion is maximum at
- A.The mean position
- B.The extreme positions
- C.One fourth of the amplitude
- D.It is constant throughout
Show answer
In simple harmonic motion the acceleration equals minus omega squared times the displacement, so it grows with displacement and is greatest at the ends of the path, where its value is A omega squared and it points back towards the centre.
Read the full explanation →Who built the first pendulum clock, using the principle discovered by Galileo?
- A.Isaac Newton
- B.Christiaan Huygens
- C.Robert Hooke
- D.Johannes Kepler
Show answer
The Dutch scientist Huygens designed the first working pendulum clock in 1656, after Galileo had noticed that a swinging lamp took the same time for each swing whether it swung far or little, the property called isochronism.
Read the full explanation →Soldiers are asked to break step while marching across a bridge in order to avoid
- A.Damping
- B.Resonance
- C.Interference
- D.Refraction
Show answer
If the frequency of the soldiers' rhythmic steps happens to match the natural frequency of the bridge, the bridge is driven at its own frequency and its amplitude of vibration can build up to a dangerous level; breaking step destroys the rhythm and avoids the risk.
Read the full explanation →Frequently Asked Questions
What is simple harmonic motion?
An oscillation in which the restoring force on the body is directly proportional to its displacement from the mean position and is always directed towards that position. A loaded spring and a pendulum swinging through a small angle are the standard examples, and the displacement varies with time as a sine function.
Why does the time period of a pendulum not depend on the mass of the bob?
Because the restoring force on the bob is itself proportional to its weight, and mass therefore cancels out when the acceleration is worked out. The period depends only on the length of the pendulum and the acceleration due to gravity, which is the law of mass.
What is a second's pendulum and how long is it?
A pendulum whose time period is exactly two seconds, so that it takes one second for each swing from one side to the other. Where the acceleration due to gravity is 9.8 metres per second squared, its length is about 99.4 centimetres, that is nearly one metre.
Why does a pendulum clock lose time in summer?
Heat makes the pendulum rod expand, so its length increases, and the time period grows as the square root of the length. Each swing takes a little longer and the clock runs slow; in winter it contracts and the clock gains. Invar, which expands very little, is used to reduce this error.
What is resonance and where is it seen?
It is the very large increase in amplitude that occurs when a body is driven by a periodic force whose frequency equals the body's own natural frequency. It is why soldiers break step on a bridge, why a tuned radio circuit picks one station, why a glass can break at a particular note and why instruments have sound boxes.
