SHM Equations

Overview

This lesson turns the definition of SHM into equations and graphs. You use the equations to calculate displacement, velocity, and acceleration, then connect those results to the shape and phase of the graphs.

What You Need to Know

• The defining equation is a = -omega^2 x.
• A common displacement solution is x = x0 sin omega t.
• Velocity can be written using v = v0 cos omega t or v = +/- omega sqrt(x0^2 - x^2).
• Displacement, velocity, and acceleration graphs are linked by gradient and phase.

How to Work Through It

1. Start by identifying amplitude, angular frequency, and the reference time in each question.
2. Choose the equation that matches the information given.
3. Track signs carefully when displacement or acceleration can be positive or negative.
4. Compare calculated values with the expected graph position.

Check Your Understanding

• Why does the acceleration have the opposite sign to the displacement?
• At which positions is speed maximum and zero?
• How are the displacement and acceleration graphs related?

Common Mistakes

• Dropping the negative sign in a = -omega^2 x when explaining direction.
• Treating x0 as a changing displacement instead of the amplitude.
• Forgetting that the +/- in the velocity equation depends on direction of motion.

Next Steps

• Practise one calculation from each equation form.
• Revisit graph interpretation before linking SHM to energy and damping.