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6.1 Period of Simple Harmonic Oscillators

6 min readapril 13, 2023

Kanya Shah

Kanya Shah

Daniella Garcia-Loos

Daniella Garcia-Loos

Kanya Shah

Kanya Shah

Daniella Garcia-Loos

Daniella Garcia-Loos

exists whenever an object is being pulled towards an equilibrium point by a force that is proportional to the displacement from the equilibrium point. Two common examples of SHM are masses on a spring (one that obeys ) and a (with a small angle displacement)

Newton's Second Law

Classically, the of an object interacting with other objects can be predicted using F = ma.

To use to solve a problem, you can follow these steps:

  1. Identify the forces acting on the object in the system. These could include , , , and any other .

  2. Draw a to represent the forces acting on the object.

  3. Determine the mass of the object and the it is experiencing.

  4. Use to write an equation for the sum of the forces acting on the object. This equation is given by: F = ma, where F is the sum of the forces, m is the mass of the object, and a is the .

  5. Substitute the known forces and the mass of the object into the equation and solve for the .

  6. Use the to determine the and position of the object as a function of time, using equations such as v = at and x = at^2/2.

  7. Graph the and position as a function of time to visualize the of the object.

  8. If necessary, use the equations for and position to solve for any unknown quantities, such as the or the initial displacement of the object.

  9. Repeat the process for any additional objects in the system, if applicable.

What is a Restoring Force?

Restoring forces can result in . When a linear is exerted on an object displaced from an , the object will undergo a special type of motion called

Here are some key points about restoring forces:

  • A is a force that acts to bring an object back to its or to maintain its .

  • Restoring forces are often encountered in systems that experience , such as a swinging back and forth or a oscillating up and down.

  • Restoring forces are often described as being "opposite in direction" to the displacement of the object from its . For example, if a is displaced to the right of its , the will be to the left, and vice versa.

  • Restoring forces can be caused by various factors, such as , , and .

  • The strength of a is often described by a , which is a measure of the stiffness of the spring or other force-generating element in the system. The higher the , the stronger the will be.

  • The can be calculated using the formula: F = -kx, where F is the , k is the , and x is the displacement of the object from its .

  • The presence of a is often used to explain why certain systems exhibit , such as a oscillating up and down or a swinging back and forth.

** Note - for AP 1, we can assume that all the springs used are ideal springs. If you plan on taking the AP C: Mech exam, that will not be the case

Periods of a Pendulum and Spring

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Fgraph.PNG?alt=media&token=c9404ce7-14bc-4ad5-9168-04af4dbc7fad

is the height of the motion measured from the equilibrium point.

is the time that it takes for an object to complete one full cycle of its motion. The is measured in seconds and is the inverse of the (measured in Hz).

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Ft.PNG?alt=media&token=f0d99ef8-3374-45cf-ba8d-4713eeb6338a

In the above equation, 𝜔 is the of the object. This will be covered in detail in Unit 7: Torque & Rotational Motion

For a , the of the oscillation can be described using the equation: 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Ftp.PNG?alt=media&token=8a37fd71-7810-4b31-b8f0-2c2e3fa15672

Where L is the length of the , and g is the .

Looking at the equation, we can see that the is proportional to the square root of the length. So a shorter will have a shorter , and vice versa. In fact in order to double the , we’d need to quadruple the length.

The for a has a very similar equation. The only main difference is that the spring’s doesn’t depend on length and , but rather the mass hung on the spring and the . For a more in-depth derivation, check out this link.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Ftp2.PNG?alt=media&token=9d92a366-3a72-43fb-a8a8-2cbc5fef0be7

Key Features of Pendulums & Springs

When dealing with pendulums and springs, a lot of the questions you’ll be dealing with refer to the velocities, forces, and accelerations at various locations in the oscillation:

  • The net force and vectors always point in the same direction. 

  • The force and vectors are greatest when the spring is fully stretched and compressed

  • The vector is 0 at the extremes where the spring is fully stretched and compressed.

  • The vector is at its maximum when the mass passes through the equilibrium point. (This is also where the force and vectors are 0)

Example Problem 1:

A mass of 1 kg is attached to a spring with a of 50 N/m and is allowed to oscillate vertically in a frictionless environment. The mass is initially displaced 0.2 meters from its and released from rest. What is the of the oscillation?

Solution:

The of an oscillation is the time it takes for the oscillating object to complete one full oscillation. For a simple , the is given by the formula: T = 2pisqrt(m/k), where T is the , m is the mass of the object, k is the , and pi is a constant equal to 3.14.

In this problem, the mass of the object is 1 kg, the is 50 N/m, and pi is 3.14.

Therefore, the of the oscillation is: T = 2(3.14)sqrt(1 kg / 50 N/m) = 0.89 seconds

This means that the of the oscillation is 0.89 seconds.

Example Problem 2:

A mass of 2 kg is attached to a spring with a of 100 N/m and is allowed to oscillate vertically in a frictionless environment. The mass is initially displaced 0.5 meters from its and released from rest. What is the of the oscillation?

Solution:

The of an oscillation is the time it takes for the oscillating object to complete one full oscillation. For a simple , the is given by the formula: T = 2pisqrt(m/k), where T is the , m is the mass of the object, k is the , and pi is a constant equal to 3.14.

In this problem, the mass of the object is 2 kg, the is 100 N/m, and pi is 3.14.

Therefore, the of the oscillation is: T = 2(3.14)sqrt(2 kg / 100 N/m) = 0.89 seconds

This means that the of the oscillation is 0.89 seconds.

🎥Watch: AP Physics 1 - Problem Solving q +a Simple Harmonic Oscillators

Key Terms to Review (27)

Acceleration

: Acceleration refers to the rate at which an object's velocity changes over time. It can be positive (speeding up), negative (slowing down), or zero (constant speed).

Acceleration due to gravity

: The acceleration due to gravity is the rate at which an object falls towards the Earth under the influence of gravity. It is approximately 9.8 meters per second squared (m/s^2) near the surface of the Earth.

Amplitude

: The amplitude represents the maximum displacement from equilibrium in a periodic motion.

Angular Velocity

: Angular velocity refers to the rate at which an object rotates or moves in a circular path. It is measured in radians per second (rad/s).

Elastic Forces

: Elastic forces are forces exerted by materials that can be stretched or compressed and return to their original shape when released. These forces are responsible for things like springs, rubber bands, and bungee cords.

Equilibrium position

: The equilibrium position is the stable, balanced point where an object or system experiences no net force and remains at rest or in uniform motion.

External Forces

: External forces are forces acting on an object that originate from outside the system being analyzed. These forces can cause changes in the motion or shape of the object.

Free-Body Diagram

: A free-body diagram is a visual representation that shows all the forces acting on an object. It helps analyze and understand how different forces affect the motion of an object.

Frequency

: Frequency is the number of complete cycles or oscillations that occur in one second. It affects how many times an object vibrates back and forth within a given time period.

Friction

: Friction is a force that opposes relative motion between two surfaces in contact. It arises due to microscopic irregularities between surfaces and can cause objects to slow down or come to rest.

Gravity

: Gravity is a fundamental force that attracts objects with mass towards each other. It is responsible for phenomena such as planetary motion and keeping us grounded on Earth.

Harmonic Oscillator

: A harmonic oscillator refers to any system where a restoring force acts on an object, causing it to oscillate back and forth around a stable equilibrium position.

Hooke’s Law

: Hooke's Law states that within the elastic limit, the force required to stretch or compress a spring by a distance is directly proportional to that distance.

Length of Pendulum

: The length of a pendulum is the distance from the pivot point to the center of mass of the object that swings back and forth.

Mass on a spring

: A mass on a spring refers to a system where a mass is attached to the end of a spring and can oscillate back and forth.

Newton's Second Law

: States that when a net external force acts on an object, the object will accelerate in the direction of the force. The acceleration is directly proportional to the net force and inversely proportional to the mass of the object.

Oscillatory Motion

: Oscillatory motion refers to the back-and-forth movement of an object around a central position, where it repeatedly passes through the same points in its path.

Pendulum

: A pendulum refers to an object suspended from a fixed point that swings freely back and forth under gravity's influence, following an arc-shaped path.

Period

: The period refers to the time it takes for one complete cycle of a repeating event or motion.

Periodic motion

: Periodic motion refers to the repetitive back-and-forth movement of an object or system in a regular pattern over time.

Position Function of Time

: The position function of time refers to a mathematical representation that describes the position of an object as a function of time. It shows how the object's position changes over time.

Restoring Force

: The restoring force refers to a force that acts on an object, pulling it back towards its equilibrium position when it is displaced from that position. It is responsible for bringing objects back to their original state.

Rotational Motion

: Rotational motion refers to the movement of an object around an axis or center point. It involves the rotation of an object rather than its linear motion and is described by quantities such as angular displacement, angular velocity, and angular acceleration.

Simple Harmonic Motion

: Simple Harmonic Motion refers to the repetitive back-and-forth motion of an object about a stable equilibrium position, where the restoring force is directly proportional to the displacement from the equilibrium position.

Spring Constant

: The spring constant represents how stiff or flexible a spring is. It determines how much force will be required to stretch or compress a spring by a certain distance.

Torque

: Torque refers to the measure of how effectively a force can cause an object to rotate around a fixed axis. It depends on both the magnitude and direction of the applied force.

Velocity

: Velocity refers to the rate at which an object changes its position in a specific direction. It includes both speed and direction.

6.1 Period of Simple Harmonic Oscillators

6 min readapril 13, 2023

Kanya Shah

Kanya Shah

Daniella Garcia-Loos

Daniella Garcia-Loos

Kanya Shah

Kanya Shah

Daniella Garcia-Loos

Daniella Garcia-Loos

exists whenever an object is being pulled towards an equilibrium point by a force that is proportional to the displacement from the equilibrium point. Two common examples of SHM are masses on a spring (one that obeys ) and a (with a small angle displacement)

Newton's Second Law

Classically, the of an object interacting with other objects can be predicted using F = ma.

To use to solve a problem, you can follow these steps:

  1. Identify the forces acting on the object in the system. These could include , , , and any other .

  2. Draw a to represent the forces acting on the object.

  3. Determine the mass of the object and the it is experiencing.

  4. Use to write an equation for the sum of the forces acting on the object. This equation is given by: F = ma, where F is the sum of the forces, m is the mass of the object, and a is the .

  5. Substitute the known forces and the mass of the object into the equation and solve for the .

  6. Use the to determine the and position of the object as a function of time, using equations such as v = at and x = at^2/2.

  7. Graph the and position as a function of time to visualize the of the object.

  8. If necessary, use the equations for and position to solve for any unknown quantities, such as the or the initial displacement of the object.

  9. Repeat the process for any additional objects in the system, if applicable.

What is a Restoring Force?

Restoring forces can result in . When a linear is exerted on an object displaced from an , the object will undergo a special type of motion called

Here are some key points about restoring forces:

  • A is a force that acts to bring an object back to its or to maintain its .

  • Restoring forces are often encountered in systems that experience , such as a swinging back and forth or a oscillating up and down.

  • Restoring forces are often described as being "opposite in direction" to the displacement of the object from its . For example, if a is displaced to the right of its , the will be to the left, and vice versa.

  • Restoring forces can be caused by various factors, such as , , and .

  • The strength of a is often described by a , which is a measure of the stiffness of the spring or other force-generating element in the system. The higher the , the stronger the will be.

  • The can be calculated using the formula: F = -kx, where F is the , k is the , and x is the displacement of the object from its .

  • The presence of a is often used to explain why certain systems exhibit , such as a oscillating up and down or a swinging back and forth.

** Note - for AP 1, we can assume that all the springs used are ideal springs. If you plan on taking the AP C: Mech exam, that will not be the case

Periods of a Pendulum and Spring

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Fgraph.PNG?alt=media&token=c9404ce7-14bc-4ad5-9168-04af4dbc7fad

is the height of the motion measured from the equilibrium point.

is the time that it takes for an object to complete one full cycle of its motion. The is measured in seconds and is the inverse of the (measured in Hz).

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Ft.PNG?alt=media&token=f0d99ef8-3374-45cf-ba8d-4713eeb6338a

In the above equation, 𝜔 is the of the object. This will be covered in detail in Unit 7: Torque & Rotational Motion

For a , the of the oscillation can be described using the equation: 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Ftp.PNG?alt=media&token=8a37fd71-7810-4b31-b8f0-2c2e3fa15672

Where L is the length of the , and g is the .

Looking at the equation, we can see that the is proportional to the square root of the length. So a shorter will have a shorter , and vice versa. In fact in order to double the , we’d need to quadruple the length.

The for a has a very similar equation. The only main difference is that the spring’s doesn’t depend on length and , but rather the mass hung on the spring and the . For a more in-depth derivation, check out this link.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Ftp2.PNG?alt=media&token=9d92a366-3a72-43fb-a8a8-2cbc5fef0be7

Key Features of Pendulums & Springs

When dealing with pendulums and springs, a lot of the questions you’ll be dealing with refer to the velocities, forces, and accelerations at various locations in the oscillation:

  • The net force and vectors always point in the same direction. 

  • The force and vectors are greatest when the spring is fully stretched and compressed

  • The vector is 0 at the extremes where the spring is fully stretched and compressed.

  • The vector is at its maximum when the mass passes through the equilibrium point. (This is also where the force and vectors are 0)

Example Problem 1:

A mass of 1 kg is attached to a spring with a of 50 N/m and is allowed to oscillate vertically in a frictionless environment. The mass is initially displaced 0.2 meters from its and released from rest. What is the of the oscillation?

Solution:

The of an oscillation is the time it takes for the oscillating object to complete one full oscillation. For a simple , the is given by the formula: T = 2pisqrt(m/k), where T is the , m is the mass of the object, k is the , and pi is a constant equal to 3.14.

In this problem, the mass of the object is 1 kg, the is 50 N/m, and pi is 3.14.

Therefore, the of the oscillation is: T = 2(3.14)sqrt(1 kg / 50 N/m) = 0.89 seconds

This means that the of the oscillation is 0.89 seconds.

Example Problem 2:

A mass of 2 kg is attached to a spring with a of 100 N/m and is allowed to oscillate vertically in a frictionless environment. The mass is initially displaced 0.5 meters from its and released from rest. What is the of the oscillation?

Solution:

The of an oscillation is the time it takes for the oscillating object to complete one full oscillation. For a simple , the is given by the formula: T = 2pisqrt(m/k), where T is the , m is the mass of the object, k is the , and pi is a constant equal to 3.14.

In this problem, the mass of the object is 2 kg, the is 100 N/m, and pi is 3.14.

Therefore, the of the oscillation is: T = 2(3.14)sqrt(2 kg / 100 N/m) = 0.89 seconds

This means that the of the oscillation is 0.89 seconds.

🎥Watch: AP Physics 1 - Problem Solving q +a Simple Harmonic Oscillators

Key Terms to Review (27)

Acceleration

: Acceleration refers to the rate at which an object's velocity changes over time. It can be positive (speeding up), negative (slowing down), or zero (constant speed).

Acceleration due to gravity

: The acceleration due to gravity is the rate at which an object falls towards the Earth under the influence of gravity. It is approximately 9.8 meters per second squared (m/s^2) near the surface of the Earth.

Amplitude

: The amplitude represents the maximum displacement from equilibrium in a periodic motion.

Angular Velocity

: Angular velocity refers to the rate at which an object rotates or moves in a circular path. It is measured in radians per second (rad/s).

Elastic Forces

: Elastic forces are forces exerted by materials that can be stretched or compressed and return to their original shape when released. These forces are responsible for things like springs, rubber bands, and bungee cords.

Equilibrium position

: The equilibrium position is the stable, balanced point where an object or system experiences no net force and remains at rest or in uniform motion.

External Forces

: External forces are forces acting on an object that originate from outside the system being analyzed. These forces can cause changes in the motion or shape of the object.

Free-Body Diagram

: A free-body diagram is a visual representation that shows all the forces acting on an object. It helps analyze and understand how different forces affect the motion of an object.

Frequency

: Frequency is the number of complete cycles or oscillations that occur in one second. It affects how many times an object vibrates back and forth within a given time period.

Friction

: Friction is a force that opposes relative motion between two surfaces in contact. It arises due to microscopic irregularities between surfaces and can cause objects to slow down or come to rest.

Gravity

: Gravity is a fundamental force that attracts objects with mass towards each other. It is responsible for phenomena such as planetary motion and keeping us grounded on Earth.

Harmonic Oscillator

: A harmonic oscillator refers to any system where a restoring force acts on an object, causing it to oscillate back and forth around a stable equilibrium position.

Hooke’s Law

: Hooke's Law states that within the elastic limit, the force required to stretch or compress a spring by a distance is directly proportional to that distance.

Length of Pendulum

: The length of a pendulum is the distance from the pivot point to the center of mass of the object that swings back and forth.

Mass on a spring

: A mass on a spring refers to a system where a mass is attached to the end of a spring and can oscillate back and forth.

Newton's Second Law

: States that when a net external force acts on an object, the object will accelerate in the direction of the force. The acceleration is directly proportional to the net force and inversely proportional to the mass of the object.

Oscillatory Motion

: Oscillatory motion refers to the back-and-forth movement of an object around a central position, where it repeatedly passes through the same points in its path.

Pendulum

: A pendulum refers to an object suspended from a fixed point that swings freely back and forth under gravity's influence, following an arc-shaped path.

Period

: The period refers to the time it takes for one complete cycle of a repeating event or motion.

Periodic motion

: Periodic motion refers to the repetitive back-and-forth movement of an object or system in a regular pattern over time.

Position Function of Time

: The position function of time refers to a mathematical representation that describes the position of an object as a function of time. It shows how the object's position changes over time.

Restoring Force

: The restoring force refers to a force that acts on an object, pulling it back towards its equilibrium position when it is displaced from that position. It is responsible for bringing objects back to their original state.

Rotational Motion

: Rotational motion refers to the movement of an object around an axis or center point. It involves the rotation of an object rather than its linear motion and is described by quantities such as angular displacement, angular velocity, and angular acceleration.

Simple Harmonic Motion

: Simple Harmonic Motion refers to the repetitive back-and-forth motion of an object about a stable equilibrium position, where the restoring force is directly proportional to the displacement from the equilibrium position.

Spring Constant

: The spring constant represents how stiff or flexible a spring is. It determines how much force will be required to stretch or compress a spring by a certain distance.

Torque

: Torque refers to the measure of how effectively a force can cause an object to rotate around a fixed axis. It depends on both the magnitude and direction of the applied force.

Velocity

: Velocity refers to the rate at which an object changes its position in a specific direction. It includes both speed and direction.


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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.


© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.