Simple Harmonic Motion

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课程笔记

Describing Oscillations

An oscillation is the repetitive variation with time of the displacement of an object about an equilibrium position.
The equilibrium position (x = 0) is where there is no resultant force on the object.
Displacement (x) is the distance from the equilibrium position; it is a vector measured in metres (m).
Amplitude (x₀) is the maximum displacement on either side of the equilibrium position, measured in metres (m).
The time period (T) is the time for one complete oscillation, measured in seconds (s).
Frequency (f) is the number of oscillations per second, measured in hertz (Hz); f = 1/T.
Angular frequency (ω) is the rate of change of angular displacement, measured in rad s⁻¹; ω = 2π/T = 2πf.

Conditions for Simple Harmonic Motion

Simple harmonic motion (SHM) is a specific type of oscillation with repetitive back-and-forth motion through an equilibrium position.
The restoring force is always directed towards the equilibrium position and is directly proportional to the displacement.
Acceleration is proportional to displacement and in the opposite direction: a ∝ −x.
The time period of oscillation is independent of the amplitude for small angles of oscillation (isochronous).
Examples of SHM include a simple pendulum, a mass on a spring, and a vibrating guitar string.
A person jumping on a trampoline is not SHM because the restoring force is not proportional to displacement and is constant when not in contact.

Defining Equation of SHM

The defining equation of SHM is a = −ω²x, where a is acceleration (m s⁻²), ω is angular frequency (rad s⁻¹), and x is displacement (m).
The minus sign indicates that acceleration is always in the opposite direction to displacement.
Acceleration is maximum when displacement is maximum (at the amplitude).
Acceleration is zero when displacement is zero (at the equilibrium position).
Velocity is maximum at the equilibrium position and zero at the amplitude positions.

Graphical Representation of SHM

Displacement, velocity, and acceleration graphs are all 90° out of phase with each other.
If oscillation starts at equilibrium, the displacement-time graph is a sine curve.
If oscillation starts at maximum displacement (amplitude), the displacement-time graph is a cosine curve.
Velocity is the gradient of the displacement-time graph; acceleration is the gradient of the velocity-time graph.
The acceleration-time graph is a reflection of the displacement-time graph in the time axis (a = −ω²x).
The amplitude is the maximum value of displacement, and the time period is the time for one full cycle.

Time Period of a Mass–Spring System

A mass-spring system consists of a mass attached to a spring; the restoring force is F = −kx.
The time period is given by T = 2π√(m/k), where m is mass (kg) and k is the spring constant (N m⁻¹).
The time period is independent of the force of gravity, so it is the same on Earth and the Moon.
A higher spring constant (stiffer spring) gives a shorter time period.
The equation applies to both horizontal and vertical mass-spring systems.
The frequency is f = 1/(2π) √(k/m).

Time Period of a Simple Pendulum

A simple pendulum consists of a point mass (bob) attached to a light, inextensible string fixed at a pivot.
The time period is given by T = 2π√(L/g), where L is the length (m) and g is gravitational field strength (N kg⁻¹).
The time period depends on gravitational field strength, so it differs on Earth and the Moon.
The formula is valid only for small angles of oscillation (θ < 10°), using the small angle approximation sin θ ≈ θ.
The restoring force is the component of weight acting along the arc towards the equilibrium position.
Angular frequency is ω = √(g/L).

Energy Changes in SHM

In SHM, energy is transferred between potential and kinetic energy stores.
For a horizontal mass-spring system, elastic potential energy is maximum at the amplitude and kinetic energy is maximum at the equilibrium position.
For a simple pendulum, gravitational potential energy is maximum at the amplitude and kinetic energy is maximum at the equilibrium position.
The total energy of an SHM system remains constant: E = Eₖ + Eₚ.
The total energy is represented by a horizontal straight line on an energy-displacement or energy-time graph.
Energy is always positive, so energy graphs have no negative values on the y-axis.

Energy–Displacement and Energy–Time Graphs

On an energy-displacement graph, potential energy is a 'U' shaped curve with maximum at amplitude and zero at equilibrium.
Kinetic energy is an 'n' shaped curve with maximum at equilibrium and zero at amplitude.
The total energy is a horizontal line above the curves.
On an energy-time graph for a pendulum, kinetic and gravitational potential energy vary periodically and in opposite directions.
When gravitational potential energy is zero, kinetic energy is maximum, and vice versa.
The kinetic and potential energy transfers go through two complete cycles during one period of oscillation.

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练习题

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1.Which of the following is the correct definition of an oscillation?
Easy
- AThe repetitive variation with time of the displacement of an object about the equilibrium position
- BThe maximum displacement of an object from its equilibrium position
- CThe time taken for one complete oscillation
- DThe number of oscillations per second
2.Which of the following statements about the equilibrium position of an oscillating object is correct?
Easy
- AIt is the position where the resultant force on the object is zero
- BIt is the position of maximum displacement
- CIt is the position where the velocity of the object is zero
- DIt is the position where the acceleration is maximum
3.Amplitude is a vector quantity.
Easy
True or false?
4.A particle oscillates with simple harmonic motion. Which of the following graphs correctly shows how the acceleration a of the particle varies with its displacement x?
Easy
- AA straight line through the origin with a negative gradient
- BA straight line through the origin with a positive gradient
- CA parabola opening upwards
- DA sine curve
5.A mass-spring system oscillates with a period T. If the spring constant is increased to 4k, what is the new period?
Easy
- AT/2
- BT/4
- C2T
- D4T
6.A simple pendulum has a length L and oscillates with a period T. If the length is reduced by 10%, what is the new period?
Easy
- A0.9T
- B0.95T
- C0.1T
7.A simple pendulum undergoes simple harmonic motion. How many times during one complete oscillation are the kinetic energy and gravitational potential energy equal?
Medium
- A1
- B2
- C3
- D4
8.Which of the following statements about the total energy of a system in simple harmonic motion is correct?
Medium
- ATotal energy = kinetic energy − potential energy
- BTotal energy = kinetic energy × potential energy
- CTotal energy = kinetic energy / potential energy
- DTotal energy = kinetic energy + potential energy

更多题目

9.Which of the following is NOT a condition for simple harmonic motion?
Medium
- AThe restoring force is directed towards the amplitude position
- BAcceleration is directed towards the equilibrium position
- CThe oscillations are isochronous
- DThe magnitude of the restoring force is proportional to the displacement
10.A point on a guitar string oscillates about its equilibrium position Z in simple harmonic motion. The amplitude of the oscillation is YZ. At which positions are the acceleration of the point at a maximum and the velocity at zero?
Medium
- AAcceleration at Z, velocity at Y
- BAcceleration at Y, velocity at Y
- CAcceleration at Z, velocity at Z
- DAcceleration at Y, velocity at Z
11.A simple pendulum and a mass-spring system oscillate about their equilibrium positions with simple harmonic motion. On Earth, the period of the oscillations is T. The pendulum and the mass-spring system are taken to Mars where the acceleration of free fall is smaller than on Earth. Which answer best describes the period of the pendulum and the mass-spring system on Mars?
Medium
- ASimple Pendulum: T, Mass-spring System: Greater than T
- BSimple Pendulum: T, Mass-spring System: T
- CSimple Pendulum: Greater than T, Mass-spring System: Greater than T
- DSimple Pendulum: Greater than T, Mass-spring System: T
12.Which of the following quantities remain constant to meet the criteria for an object oscillating in simple harmonic motion?
Medium
- AFrequency, f
- BFrequency, f, and period, T
- CPeriod, T, and the spring constant, k
- DPeriod, T, frequency, f, spring constant, k, and acceleration of free fall, g
13.Which of the following is a correct arrangement for the maximum displacement x₀ of a particle performing simple harmonic motion?
Medium
- Ax₀ = −(a_max f²)/(4π²)
- Bx₀ = −(a_max)/(2π f²)
- Cx₀ = −(a_max)/(4π² T²)
- Dx₀ = −(a_max T²)/(4π²)
14.A mass-spring system oscillates about its equilibrium position in simple harmonic motion. A bulb on the motion sensor lights up each time the block passes the equilibrium position. The block has a mass m and oscillates with a period T. Select the new mass that would cause the period to double.
Medium
- A0.5m
- B1.4m
- C2m
- D4m
15.A particle, P, oscillates on the line XZ about its equilibrium point Y, in simple harmonic motion. At the point shown, which statement could be correct about the motion of the particle?
Hard
- AIt has maximum kinetic energy and minimum potential energy
- BThe total energy is equal to the kinetic energy at X
- CThe restoring force is towards Z and the particle is accelerating
- DThe restoring force is towards X and the particle is accelerating
16.A pendulum is undergoing simple harmonic motion with a time period T and angular frequency ω. A student makes a change to the set up so that the pendulum has a new time period 3T. What is the new angular frequency?
Hard
- A6ω
- B3ω
- Cω/3
- Dω/6
17.A simple pendulum oscillates with simple harmonic motion. At which positions are the acceleration at zero, the displacement at a negative maximum, and velocity at a maximum?
Hard
- AAcceleration at Z, displacement at Y, velocity at X
- BAcceleration at Y, displacement at X, velocity at Y
- CAcceleration at X, displacement at Z, velocity at Z
- DAcceleration at Y, displacement at X, velocity at Z
18.Which of the following statements about simple harmonic motion are correct? (Select all that apply)
Medium
- AThe acceleration is proportional to the displacement
- BThe acceleration is in the opposite direction to the displacement
- CThe time period is independent of the amplitude for small angles
- DThe restoring force is always directed towards the amplitude position
- EThe velocity is maximum at the amplitude position
19.Which of the following are examples of simple harmonic motion? (Select all that apply)
Hard
- AA child on a swing
- BA mass on a spring
- CA person jumping on a trampoline
- DA guitar string vibrating
- EA marble rolling in a bowl
20.The time period of a simple pendulum depends on the mass of the bob.
Easy
True or false?
21.In simple harmonic motion, the total energy of the system remains constant.
Easy
True or false?
22.Match each quantity with its correct SI unit.
Medium
- Displacement
- Amplitude
- Time period
- Frequency
- Angular frequency
- m
- s
- Hz
- rad s⁻¹
23.Place the following steps in the correct order to determine the angular frequency of a cuckoo emerging from a cuckoo clock, given it travels from fully compressed to fully extended in 1.5 s.
Medium
- Recognise the angular displacement is π radians
- Use the equation ω = 2π/T = π/1.5
- Calculate ω = 2.09 rad s⁻¹
- State the final answer as 2.1 rad s⁻¹ (2 s.f.)
24.A mass-spring system oscillates in simple harmonic motion. The mass is 0.50 kg and the spring constant is 2.0 N m⁻¹. What is the time period of the oscillation?
Easy
- A0.50 s
- B1.0 s
- C3.1 s
- D6.3 s
25.A simple pendulum has a length L and oscillates with a time period T. The length is increased to 4L. What is the new time period?
Medium
- A0.5T
- B2T
- C4T
- D16T
26.A particle oscillates with simple harmonic motion. The defining equation is a = −ω²x. If the angular frequency ω is doubled, by what factor does the magnitude of the acceleration change for the same displacement?
Easy
- AIt halves
- BIt stays the same
- CIt doubles
- DIt quadruples
27.A simple pendulum oscillates with simple harmonic motion. At which position is the kinetic energy of the pendulum at a maximum?
Medium
- AAt the amplitude position
- BAt the equilibrium position
- CHalfway between the equilibrium and amplitude positions
- DThe kinetic energy is constant throughout the oscillation
28.Which of the following statements are true for an object undergoing simple harmonic motion? (Select all that apply)
Medium
- AThe acceleration is proportional to the displacement from equilibrium.
- BThe acceleration is always directed towards the equilibrium position.
- CThe velocity is always in the same direction as the displacement.
- DThe time period is independent of the amplitude for small oscillations.
- EThe total energy of the system decreases as the object moves away from equilibrium.
29.A particle oscillates with simple harmonic motion. Its displacement–time graph starts at the amplitude position at t = 0. Which of the following correctly describes the shapes of the velocity–time graph and the acceleration–time graph?
Medium
- AVelocity: negative sine curve; Acceleration: negative cosine curve
- BVelocity: cosine curve; Acceleration: negative sine curve
- CVelocity: negative cosine curve; Acceleration: sine curve
- DVelocity: sine curve; Acceleration: cosine curve
30.A mass–spring system has a period T. The mass is changed to 4m while the spring constant stays the same. What is the new period?
Medium
- A2T
- B4T
- CT/2
- DT/4
31.A pendulum of length L has a period T on Earth. It is taken to the Moon where the gravitational field strength is g/6. What is the period of the pendulum on the Moon?
Medium
- A2.45T
- B0.41T
- C6T
- DT
32.The graph of kinetic energy against displacement for a horizontal mass–spring system is an 'n'-shaped curve. What is the shape of the potential energy against displacement graph for the same system?
Medium
- AA 'U'-shaped curve
- BA horizontal straight line
- CAn 'n'-shaped curve
- DA straight line through the origin
33.A particle in simple harmonic motion has an angular frequency of 4.0 rad s⁻¹ and an amplitude of 0.10 m. What is its maximum acceleration?
Medium
- A1.6 m s⁻²
- B0.40 m s⁻²
- C0.016 m s⁻²
- D4.0 m s⁻²
34.A simple pendulum oscillates with simple harmonic motion. Which of the following statements are correct? (Select all that apply)
Medium
- AAt the equilibrium position, kinetic energy is maximum and gravitational potential energy is minimum
- BAt the amplitude positions, kinetic energy is zero and gravitational potential energy is maximum
- CThe total energy of the pendulum remains constant throughout the oscillation
- DThe period of the pendulum depends on the mass of the bob
- EThe period of the pendulum depends on the gravitational field strength
35.A particle oscillates in simple harmonic motion. Which of the following are true about the phase relationships between displacement, velocity and acceleration? (Select all that apply)
Medium
- AVelocity leads displacement by 90°
- BAcceleration is 180° out of phase with displacement
- CVelocity and acceleration are in phase with each other
- DAcceleration leads velocity by 90°
- EDisplacement and velocity are in phase with each other
36.The time period of a simple pendulum is independent of the mass of the bob.
Easy
True or false?
37.In simple harmonic motion, the acceleration is always in the same direction as the displacement.
Medium
True or false?
38.The time period of a mass–spring system is the same on the Earth and on the Moon.
Medium
True or false?
39.Match each SHM quantity with its correct SI unit.
Medium
- Displacement
- Time period
- Frequency
- Angular frequency
- Spring constant
- m
- s
- Hz
- rad s⁻¹
- N m⁻¹
40.Place the following steps in the correct order to determine the maximum velocity of an object oscillating with simple harmonic motion, given its maximum kinetic energy and mass.
Medium
- Write down the known values of maximum kinetic energy and mass
- Write down the equation for kinetic energy: Ek = ½mv²
- Rearrange the equation to make velocity the subject: v = √(2Ek/m)
- Substitute the values into the rearranged equation and calculate
41.Complete the sentence about the defining equation of simple harmonic motion.
Easy
The defining equation of simple harmonic motion is a = −____ x, where ω is the ____ frequency.
42.Complete the equation for the time period of a mass-spring system.
Medium
T = 2π√( ____ / ____ )
43.A child on a swing performs 0.2 oscillations per second. Calculate the time period of the oscillation.
Medium
44.A mass of 2.0 kg is attached to a spring with a spring constant of 0.9 N m⁻¹ and oscillates with simple harmonic motion. Calculate the frequency of the oscillation.
Medium
45.What is the term for oscillations that have a constant time period?
Easy
46.What is the SI unit of angular frequency?
Medium
47.Calculate the force that acts on the robot when the spring is extended by 5 cm.
Medium
48.Calculate the acceleration of the robot whilst at its amplitude position.
Medium
49.Determine the total energy of the object.
Medium
50.Determine the maximum velocity of the object.
Medium
51.Determine the potential energy of the object when the displacement is x = 1.0 cm.
Medium
52.Calculate the force that acts on the robot when the spring is extended by 5 cm. Give your answer in N and include the sign.
Medium
53.Explain why a person jumping on a trampoline is not an example of simple harmonic motion.
Hard
54.State the two conditions required for an object to undergo simple harmonic motion.
Medium
55.A student says that a mass bouncing on a spring is an example of simple harmonic motion only if the spring obeys Hooke's Law. Explain why this is the case.
Hard
56.A child on a swing performs 0.2 oscillations per second. Calculate the time period of the oscillation.
Medium
- A4 s
- B7 s
- C5 s
- D6 s
57.A mass of 2.0 kg is attached to a spring with a spring constant of 0.9 N m⁻¹ and oscillates with simple harmonic motion. Calculate the frequency of the oscillation.
Medium
- A2.11 Hz
- B0.11 Hz
- C-0.89 Hz
- D1.11 Hz
58.Calculate the force that acts on the robot when the spring is extended by 5 cm.
Medium
- A6.5 N
- B3.5 N
- C5.5 N
- D4.5 N
59.Calculate the acceleration of the robot whilst at its amplitude position.
Medium
- A24.8 m s⁻²
- B27.1 m s⁻²
- C20.2 m s⁻²
- D22.5 m s⁻²
60.Determine the total energy of the object.
Medium
- A-0.94 J
- B1.06 J
- C0.06 J
- D2.06 J
61.Determine the maximum velocity of the object.
Medium
- A1.49 m s⁻¹
- B-0.51 m s⁻¹
- C0.49 m s⁻¹
- D2.49 m s⁻¹
62.Determine the potential energy of the object when the displacement is x = 1.0 cm.
Medium
- A0.01 J
- B-0.99 J
- C1.01 J
- D2.01 J
63.Calculate the force that acts on the robot when the spring is extended by 5 cm. Give your answer in N and include the sign.
Medium
- A-2.5 N
- B-5.5 N
- C-4.5 N
- D-3.5 N

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