Reading Time: 2 minutesDisplacement (
Velocity (
Acceleration (
Kinetic Energy (
Potential Energy (
Simple Harmonic Motion (SHM): The Science of Oscillations
What Is Simple Harmonic Motion?
SHM describes periodic oscillations where the restoring force is proportional to displacement and directed toward equilibrium.
Key Equations in SHM
Displacement (
)
)The position of the object at time 
:
![\[ x = A \cos(\omega t + \phi) \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-a8b97ffffad8729eb10f801cb9fc4dd5_l3.png "Rendered by QuickLaTeX.com")
Where:
: Amplitude
: Angular frequency (rad/s)
: Phase constant
: Amplitude
: Angular frequency (rad/s)
: Phase constantVelocity (
)
)The rate of change of displacement:
![\[ v = -\omega A \sin(\omega t + \phi) \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-fbbae7f18c7e452c037a756cfce49a5c_l3.png "Rendered by QuickLaTeX.com")
Acceleration ()
)Proportional to displacement:
![\[ a = -\omega^2 x \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-ab0391b504ca86a9405c16765bec5fbd_l3.png "Rendered by QuickLaTeX.com")
Energy in SHM
Kinetic Energy (
)
)Energy due to motion:
![\[ KE = \frac{1}{2} m \omega^2 (A^2 - x^2) \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-dc40580dc4d0c54a29b09b9c1392f252_l3.png "Rendered by QuickLaTeX.com")
Potential Energy (
)
)Energy due to position:
![\[ PE = \frac{1}{2} m \omega^2 x^2 \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-09411be49632217dce1ff932c0048385_l3.png "Rendered by QuickLaTeX.com")
Applications of SHM
Clocks
Pendulums use SHM to maintain consistent timekeeping.
Engineering
SHM principles are applied in suspension systems and shock absorbers.
Medical Imaging
Ultrasound machines rely on oscillating sound waves.
Example Problem
A 
mass oscillates on a spring with a spring constant of 
and amplitude of 
. Find the period and maximum velocity.
- Angular Frequency ():
![\[ \omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{50}{2}} = 5 \, \text{rad/s} \]](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMTMiIGhlaWdodD0iNDMiIHZpZXdCb3g9IjAgMCAyMTMgNDMiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+ "Rendered by QuickLaTeX.com")
- Period (
):![\[ T = \frac{2\pi}{\omega} = \frac{2\pi}{5} \approx 1.26 \, \text{s} \]](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNzIiIGhlaWdodD0iMzYiIHZpZXdCb3g9IjAgMCAxNzIgMzYiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+ "Rendered by QuickLaTeX.com")
- Maximum Velocity (
):![\[ v_{\text{max}} = \omega A = 5 \times 0.1 = 0.5 \, \text{m/s} \]](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTkiIHZpZXdCb3g9IjAgMCAyNDAgMTkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+ "Rendered by QuickLaTeX.com")
![\[ \omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{50}{2}} = 5 \, \text{rad/s} \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-2579d4eb7a6873d432826029047c768e_l3.png "Rendered by QuickLaTeX.com")
):![\[ T = \frac{2\pi}{\omega} = \frac{2\pi}{5} \approx 1.26 \, \text{s} \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-c0fbf058ea2383eaf6b6a032fb82d0ed_l3.png "Rendered by QuickLaTeX.com")
):![\[ v_{\text{max}} = \omega A = 5 \times 0.1 = 0.5 \, \text{m/s} \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-6c21d5dd11ba8720b3bafea90fade0f7_l3.png "Rendered by QuickLaTeX.com")
Common Mistakes in SHM Problems
- Forgetting the negative sign in the restoring force equation
- Mixing up displacement and amplitude
- Confusing angular frequency with regular frequency
Practice Questions
- A
mass oscillates with an amplitude of 
and a spring constant of 
. Find the maximum velocity. - Explain how energy is conserved in SHM.
- Describe one real-world example of SHM in engineering.
mass oscillates with an amplitude of 
and a spring constant of 
. Find the maximum velocity.