Simple Harmonic Motion

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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 acting on the object.
Displacement (x) is the distance of a point from its equilibrium position; it is a vector measured in metres (m).
Amplitude (x₀) is the maximum value of displacement on either side of the equilibrium position, measured in metres (m).
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 with respect to time, measured in rad s⁻¹; ω = 2π/T = 2πf.

Simple Harmonic Motion (SHM)

Simple harmonic motion is a specific type of oscillation where the acceleration is proportional to the displacement and is always directed towards the equilibrium position.
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 shows that acceleration and displacement are always in opposite directions.
The time period of SHM is independent of the amplitude for small angles of oscillation.
Examples of SHM include a pendulum, a mass on a spring, and a marble in a bowl.
A person jumping on a trampoline is not SHM because the restoring force is not proportional to displacement.

Graphical Representation of SHM

Displacement, velocity and acceleration against time can be represented by sine or cosine curves.
If oscillations start from the equilibrium position, displacement is a sine curve, velocity is a cosine curve, and acceleration is a negative sine curve.
If oscillations start from the amplitude position, displacement is a cosine curve, velocity is a negative sine curve, and acceleration is a negative cosine curve.
The displacement, velocity and acceleration graphs are all 90° out of phase with each other.
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 x-axis.

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 the mass (kg) and k is the spring constant (N m⁻¹).
This equation applies to both horizontal and vertical mass-spring systems.
The time period does not depend on the force of gravity, so it is the same on Earth and the Moon.
A higher spring constant k means a stiffer spring and a shorter time period.

Time Period of a Simple Pendulum

A simple pendulum consists of a point mass (bob) attached to a light, inextensible string fixed at a point above.
For small angles (θ < 10°), the time period is T = 2π√(L/g), where L is the length (m) and g is gravitational field strength (N kg⁻¹).
The small angle approximation sin θ ≈ θ is used to derive this equation.
The time period depends on gravitational field strength, so it differs on Earth and the Moon.
The time period is independent of the mass of the bob and the amplitude (for small angles).

Energy Changes in SHM

In SHM, energy is continuously transferred between potential and kinetic energy.
For a horizontal mass-spring system, elastic potential energy is maximum at the amplitude positions and zero at equilibrium.
For a simple pendulum, gravitational potential energy is maximum at the amplitude positions and zero at equilibrium.
Kinetic energy is maximum at the equilibrium position and zero at the amplitude positions.
The total energy of an SHM system remains constant, equal to the sum of kinetic and potential energy.
The total energy is given by E = ½mω²x₀², where m is mass, ω is angular frequency, and x₀ is amplitude.
The potential energy at displacement x is Eₚ = ½mω²x².
The kinetic energy at displacement x is Eₖ = ½mω²(x₀² − x²).

Equations for SHM

For oscillations starting from equilibrium (x = 0 at t = 0): x = x₀ sin ωt, v = ωx₀ cos ωt, a = −ω²x₀ sin ωt.
For oscillations starting from amplitude (x = x₀ at t = 0): x = x₀ cos ωt, v = −ωx₀ sin ωt, a = −ω²x₀ cos ωt.
The velocity-displacement relation is v = ±ω√(x₀² − x²).
Maximum velocity occurs at equilibrium and is v_max = ωx₀.
Maximum acceleration occurs at amplitude and is a_max = ω²x₀.
These equations are in the data booklet and must be used with radians mode on calculators.

Phase Angles in SHM

The phase angle φ is the difference in angular displacement compared to an oscillator with x = 0 at t = 0.
The phase angle can range from 0 to 2π radians.
With phase angle, the displacement equation becomes x = x₀ sin(ωt + φ).
A sine wave leads a cosine wave by π/2 radians, or equivalently, cosine lags sine by π/2.
For a wave that lags, the phase difference is +π/2; for a wave that leads, it is −π/2.
Sine and cosine functions are out of phase by π/2 radians.
Phase shifts: sin(ωt − φ) shifts right, sin(ωt + φ) shifts left, cos(ωt − φ) shifts right, cos(ωt + φ) shifts left.

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Übungsfragen

Gratis-Vorschau — 8 von 64 Fragen. Registriere dich, um alle zu sehen.

1.A pendulum oscillates with a time period of 2.2 s. Calculate the frequency of the oscillation.
Easy
- A0.45 Hz
- B2.2 Hz
- C4.4 Hz
- D0.22 Hz
2.A mass-spring system oscillates with simple harmonic motion. The total energy of the system is 60 mJ. What is the maximum kinetic energy of the system?
Medium
- A60 mJ
- B30 mJ
- C0 mJ
- D120 mJ
3.A mass-spring system oscillates with simple harmonic motion. The graph of potential energy against displacement shows a maximum potential energy of 40 mJ at an amplitude of 0.02 m. What is the total energy of the system?
Medium
- A40 mJ
- B20 mJ
- C80 mJ
- D0 mJ
4.A block of mass 25 g oscillates with simple harmonic motion with a maximum kinetic energy of 40 mJ. Calculate the maximum velocity of the block.
Medium
- A1.8 m/s
- B0.9 m/s
- C3.2 m/s
- D0.45 m/s
5.A spring has a spring constant of 2.1 N/m. The mass-spring system oscillates with an amplitude of 0.1 m. Calculate the restoring force at the amplitude position.
Medium
- A0.21 N
- B2.1 N
- C0.021 N
- D21 N
6.A pendulum bob oscillates with simple harmonic motion. At which position does the pendulum have maximum kinetic energy?
Medium
- AAt the equilibrium position
- BAt maximum displacement
- CAt half the amplitude
- DAt the highest point
7.The time period of a simple pendulum depends on its mass.
Easy
True or false?
8.In simple harmonic motion, the acceleration is always directed towards the equilibrium position.
Easy
True or false?

Mehr Fragen

9.Which of the following are examples of simple harmonic motion? (Select all that apply)
Medium
- AA mass on a spring
- BA simple pendulum with small oscillations
- CA person jumping on a trampoline
- DA guitar string vibrating
- EA ball rolling down a hill
10.Match the following terms with their correct definitions.
Medium
- Displacement
- Amplitude
- Time period
- Frequency
- The maximum displacement from equilibrium
- The time for one complete oscillation
- The number of oscillations per second
- The distance from equilibrium at a given time
11.Match each term with its correct definition.
Medium
- Displacement
- Amplitude
- Time period
- Frequency
- Angular frequency
- The maximum displacement from the equilibrium position
- The distance of an object from its equilibrium position at a given time
- The time taken for one complete oscillation
- The rate of change of angular displacement with respect to time
- The number of complete oscillations per second
12.For a simple pendulum swinging with simple harmonic motion, arrange the following positions in order of increasing kinetic energy, starting from the lowest.
Medium
- At the amplitude position (maximum displacement)
- Halfway between amplitude and equilibrium
- At the equilibrium position
13.A mass-spring system has a time period of 2.0 s. What is its angular frequency?
Medium
- Aπ rad s⁻¹
- B2π rad s⁻¹
- C0.32 rad s⁻¹
- D4π rad s⁻¹
14.A simple pendulum has a time period T. If its length is quadrupled, what is the new time period?
Hard
- A2T
- B4T
- CT/2
- DT/4
15.Which of the following statements about the energy of a simple harmonic oscillator are correct? (Select all that apply)
Hard
- AThe total energy is constant if there is no damping.
- BPotential energy is maximum at the equilibrium position.
- CKinetic energy is maximum at the equilibrium position.
- DTotal energy is proportional to the square of the amplitude.
- EPotential energy is always equal to kinetic energy.
16.A mass on a spring is pulled down and released. It oscillates with simple harmonic motion. At which point in the oscillation is the acceleration of the mass zero?
Medium
- AAt the equilibrium position
- BAt the amplitude position
- CHalfway between equilibrium and amplitude
- DThe acceleration is never zero
17.A simple pendulum oscillates with a period of 2.0 s. What is its length? (Take g = 9.8 m/s²)
Medium
- A0.99 m
- B1.0 m
- C2.0 m
- D4.0 m
18.A student investigates a mass-spring system oscillating with simple harmonic motion. Which measurement would best reduce the percentage uncertainty in the time period?
Medium
- ATime 20 complete oscillations and divide by 20
- BTime one oscillation with a stopwatch
- CMeasure the amplitude and calculate the period
- DTime the first half oscillation only
19.A simple pendulum is set up to determine g. The length L is measured from the pivot to the centre of the bob. Which change would improve the accuracy of the experiment most?
Medium
- AUse a larger amplitude of swing
- BUse a heavier bob of the same size
- CMeasure the time for 10 oscillations instead of 1
- DUse a shorter string
20.A mass-spring system oscillates vertically. The graph of displacement against time is a cosine curve starting at maximum positive displacement. Which graph represents its acceleration?
Hard
- AA negative cosine curve
- BA positive sine curve
- CA negative sine curve
- DA positive cosine curve
21.Which of the following are essential precautions when investigating the time period of a simple pendulum? (Select all that apply)
Medium
- AUse a small angle of swing (less than 10°)
- BMeasure the length from the pivot to the centre of the bob
- CTime several oscillations to reduce timing uncertainty
- DUse a heavy bob to reduce air resistance effects
- EEnsure the pendulum swings in a single plane
22.The time period of a simple pendulum is independent of its mass.
Easy
True or false?
23.Match each experimental quantity with the correct method of measurement in a mass-spring oscillation experiment.
Medium
- Time period
- Amplitude
- Spring constant
- Time 10 oscillations and divide by 10
- Measure maximum displacement from equilibrium with a ruler
- Use Hooke's Law with known masses
24.Arrange the steps to determine the spring constant k from a static mass-spring experiment in the correct order.
Medium
- Measure the initial position of the bottom of the spring
- Add a known mass and measure the new position
- Calculate the extension from the difference in positions
- Plot a graph of force against extension and find the gradient
25.What is the SI unit of frequency?
Easy
- AHertz (Hz)
- BSecond (s)
- CRadian per second (rad s⁻¹)
- DMetre (m)
26.Which of the following is the correct definition of displacement in the context of oscillations?
Easy
- AThe maximum distance from the equilibrium position
- BThe total distance travelled by the oscillating object
- CThe distance of an object from its equilibrium position, in a specified direction
- DThe time taken for one complete oscillation
27.Which of the following is the correct definition of amplitude?
Easy
- AThe maximum displacement from the equilibrium position
- BThe time taken for one complete oscillation
- CThe number of oscillations per second
- DThe distance between two consecutive peaks of an oscillation
28.Which of the following is the correct definition of time period?
Easy
- AThe time taken for half an oscillation
- BThe time taken for one complete oscillation
- CThe number of oscillations per second
- DThe maximum displacement from equilibrium
29.Which of the following equations correctly relates the time period T and frequency f of an oscillation?
Medium
- AT = 1/f
- BT = f
- CT = 2πf
- DT = 1/(2πf)
30.Which of the following expressions correctly gives the angular frequency ω in terms of the time period T?
Easy
- Aω = 2π/T
- Bω = T/(2π)
- Cω = 2πT
- Dω = 1/T
31.Which of the following is the defining equation of simple harmonic motion?
Medium
- Aa = −ω²x
- Ba = ω²x
- Ca = −ωx
- Da = ωx
32.Which of the following statements are true for an object undergoing simple harmonic motion? (Select all that apply)
Medium
- AThe acceleration is directly proportional to the displacement from equilibrium.
- BThe acceleration is always directed towards the equilibrium position.
- CThe time period depends on the amplitude for small oscillations.
- DThe velocity is maximum at the equilibrium position.
- EThe acceleration is maximum at the equilibrium position.
33.The time period of a mass-spring system depends on the force of gravity.
Easy
True or false?
34.The total energy of a simple harmonic oscillator remains constant if there is no damping.
Easy
True or false?
35.In simple harmonic motion, the velocity is maximum at the amplitude positions.
Easy
True or false?
36.The period of a simple pendulum is independent of the amplitude for small angles of oscillation.
Easy
True or false?
37.Match each quantity with its correct SI unit.
Medium
- Displacement
- Frequency
- Angular frequency
- Time period
- metre (m)
- hertz (Hz)
- radian per second (rad s⁻¹)
- second (s)
38.Match each equation with the physical quantity it defines.
Medium
- T = 1/f
- ω = 2π/T
- a = −ω²x
- T = 2π√(m/k)
- T = 2π√(L/g)
- Relationship between time period and frequency
- Angular frequency in terms of time period
- Defining equation of simple harmonic motion
- Time period of a mass-spring system
- Time period of a simple pendulum
39.Arrange the following steps in the correct order to describe the energy changes during one complete oscillation of a simple pendulum, starting from the amplitude position.
Easy
- Pendulum at amplitude: maximum gravitational potential energy, zero kinetic energy
- Pendulum moves towards equilibrium: gravitational potential energy decreases, kinetic energy increases
- Pendulum at equilibrium: maximum kinetic energy, minimum gravitational potential energy
- Pendulum moves away from equilibrium: kinetic energy decreases, gravitational potential energy increases
- Pendulum returns to amplitude: maximum gravitational potential energy, zero kinetic energy
40.Arrange the following steps in the correct order to derive the time period of a mass-spring system using the defining equation of SHM.
Medium
- Start with the restoring force F = −kx and Newton's second law F = ma.
- Equate to get ma = −kx, so a = −(k/m)x.
- Compare with the defining equation a = −ω²x to identify ω² = k/m.
- Solve for ω = √(k/m).
- Substitute into T = 2π/ω to get T = 2π√(m/k).
41.What is the SI unit of angular frequency?
Medium
42.Complete the sentence about the defining equation of simple harmonic motion.
Easy
For an object in simple harmonic motion, the acceleration is proportional to the ____ and is always directed ____ the equilibrium position.
43.A mass-spring system oscillates with a period of 1.84 s. Calculate the angular frequency of the oscillation. Give your answer to 3 significant figures.
Medium
44.A mass-spring system oscillates with a period of 0.50 s. Calculate the angular frequency of the oscillation.
Medium
45.Complete the definition of an oscillation.
Easy
An oscillation is the repetitive variation with time of the ____ of an object about the ____ position.
46.Complete the sentence about the conditions for simple harmonic motion.
Easy
For simple harmonic motion, the acceleration is ____ to the displacement and is in the ____ direction to the displacement.
47.A pendulum completes 10 oscillations in 12.0 s. Calculate the frequency of the oscillations. Give your answer to 2 significant figures.
Medium
48.A guitar string vibrates with simple harmonic motion at a frequency of 225 Hz. Calculate the angular frequency of the vibration. Give your answer to 3 significant figures.
Medium
49.A 55 g mass attached to a spring oscillates on a frictionless horizontal surface with a period of 1.84 s. Calculate the spring constant of the spring. Give your answer to 3 significant figures.
Medium
50.A 2750 kg steel sphere hangs from a 12 m cable and is pulled 2.0 m to the side and released, oscillating as a simple pendulum. Calculate the angular frequency of the oscillation. Take g = 9.8 m/s². Give your answer to 3 significant figures.
Medium
51.Complete the equations for the time period of these two oscillating systems. Use the symbols T, m, k, L and g.
Easy
For a mass-spring system: T = 2π√(____ / ____) For a simple pendulum: T = 2π√(____ / ____)
52.A mass-spring system has a mass of 2.0 kg and a spring constant of 0.90 N/m. Calculate the frequency of the oscillation. Give your answer to 2 significant figures.
Medium
53.Explain why a person jumping on a trampoline is not an example of simple harmonic motion.
Hard
54.Describe the conditions required for an object to perform simple harmonic motion (SHM).
Medium
55.Explain why, in reality, a freely oscillating pendulum cannot maintain a constant amplitude.
Hard
56.A smooth glass marble is held at the edge of a bowl and released, and it rolls up and down the sides of the bowl. The magnitude of the restoring force is given by F = mgx/R, where x is the displacement and R is the radius of the bowl. Outline why the oscillations can be described as simple harmonic motion.
Hard
57.Explain why the time period of a simple pendulum is independent of the mass of the bob.
Hard
58.A mass-spring system oscillates with a period of 1.84 s. Calculate the angular frequency of the oscillation. Give your answer to 3 significant figures.
Medium
- A4.41 rad s⁻¹
- B2.41 rad s⁻¹
- C5.41 rad s⁻¹
- D3.41 rad s⁻¹
59.A mass-spring system oscillates with a period of 0.50 s. Calculate the angular frequency of the oscillation.
Medium
- A14.6 rad/s
- B12.6 rad/s
- C13.6 rad/s
- D11.6 rad/s
60.A pendulum completes 10 oscillations in 12.0 s. Calculate the frequency of the oscillations. Give your answer to 2 significant figures.
Medium
- A0.83 Hz
- B2.83 Hz
- C-0.17 Hz
- D1.83 Hz
61.A guitar string vibrates with simple harmonic motion at a frequency of 225 Hz. Calculate the angular frequency of the vibration. Give your answer to 3 significant figures.
Medium
- A1551 rad/s
- B1269 rad/s
- C1692 rad/s
- D1410 rad/s
62.A 55 g mass attached to a spring oscillates on a frictionless horizontal surface with a period of 1.84 s. Calculate the spring constant of the spring. Give your answer to 3 significant figures.
Medium
- A2.641 N/m
- B1.641 N/m
- C-0.359 N/m
- D0.641 N/m
63.A 2750 kg steel sphere hangs from a 12 m cable and is pulled 2.0 m to the side and released, oscillating as a simple pendulum. Calculate the angular frequency of the oscillation. Take g = 9.8 m/s². Give your answer to 3 significant figures.
Medium
- A2.904 rad/s
- B-0.096 rad/s
- C0.904 rad/s
- D1.904 rad/s
64.A mass-spring system has a mass of 2.0 kg and a spring constant of 0.90 N/m. Calculate the frequency of the oscillation. Give your answer to 2 significant figures.
Medium
- A2.11 Hz
- B-0.89 Hz
- C0.11 Hz
- D1.11 Hz

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