PHYC10003 Chap.6 Oscillation Energy and Pendulums
Oscillation Energy and Pendulums
Ideal spring motion exchanges kinetic and potential energy while total mechanical energy remains constant. Vertical springs require displacement to be measured about the gravity-shifted equilibrium. Pendulum models then split into simple and physical cases according to mass distribution, with the small-angle approximation creating simple harmonic motion and symbolic simplification revealing cancellations.
At either spring endpoint, speed and kinetic energy vanish; at equilibrium, speed and kinetic energy are greatest. Because both energy forms contain squared quantities, equal positive and negative displacements have equal energy. Damping transfers mechanical energy away and reduces the amplitude envelope, so constant-energy shortcuts no longer apply.
A vertical spring has a static gravitational extension that locates a new equilibrium. Measuring oscillation displacement from that point removes the constant weight term from the ideal timing equation. A simple pendulum treats its mass as concentrated at one distance from the pivot. A physical pendulum uses the object's moment of inertia about the pivot and its centre-of-mass distance.
In both cases, the elementary period relation requires small angular displacement in radians so the restoring torque is approximately linear. Symbolic work often reveals mass cancellation and the correct dependence on gravity, length, or inertia. Period, frequency, and angular frequency must remain separately labelled to avoid reciprocal or unit errors.
Qualitative trends provide a final check: weaker gravity lengthens a pendulum period, and moving mass farther from the pivot can slow an extended body.
What this chapter covers
- 01
Spring potential energy
- 02
Energy exchange through a cycle
- 03
Vertical equilibrium extension
- 04
Simple pendulum
- 05
Small-angle approximation
- 06
Physical pendulum and centre of mass
Worked example · free
Simplify a uniform-rod pendulum
- 1Write angular frequency squared as mgd divided by I.
- 1Substitute d equals L over two and the supplied inertia.
- 1Cancel mass and simplify to 3g over 2L.
- 1Calculate period as 1.56 seconds.
Key terms
- Spring Constant
- A spring constant measures the stiffness relating restoring force to displacement.
- Equilibrium Extension
- Equilibrium extension is the static stretch where spring force balances weight.
- Small-angle Approximation
- The small-angle approximation replaces sine of an angle by the angle measured in radians.
- Simple Pendulum
- A simple pendulum idealises all moving mass at a point a fixed distance from the pivot.
- Physical Pendulum
- A physical pendulum accounts for distributed mass through moment of inertia and centre-of-mass distance.
Oscillation Energy and Pendulums FAQ
Where is spring energy largest during motion?
Spring potential energy is largest at either endpoint where displacement magnitude equals amplitude. Kinetic energy is then zero. At equilibrium, spring energy is minimum and kinetic energy is maximum. At intermediate positions, both forms are present and must sum to the same ideal mechanical total.
Does gravity change vertical spring frequency?
Gravity shifts the equilibrium position by a static extension. When displacement is measured from that new equilibrium, the ideal angular frequency remains the square root of spring constant divided by mass.
Why does pendulum mass often cancel?
Both gravitational restoring torque and rotational inertia scale with mass under the ideal model. Writing the symbolic ratio before numbers makes this shared factor visible and allows it to cancel.
Exam move
Draw kinetic, spring-potential, and total-energy curves against position and mark both endpoints and equilibrium. At each landmark, state displacement, speed, acceleration direction, and the form in which mechanical energy is stored. At intermediate positions, subtract potential energy from the fixed total and solve for speed, checking symmetry between positive and negative displacement.
Compare the energy curves with the time graphs: energy repeats after half a motion cycle because squared displacement and speed ignore sign. Add a damped case and explain why its amplitude envelope and mechanical total decrease even though the restoring relation can still point toward equilibrium. For vertical springs, calculate static extension separately from oscillation displacement.
Shift the coordinate origin to the hanging equilibrium before applying the ideal angular frequency so gravity is not counted twice. Compare simple and physical pendulum diagrams by locating the pivot, centre of mass, length variable, and required moment of inertia. For an extended body, write gravitational torque scale over rotational inertia symbolically and cancel shared mass factors before using numbers.
Test a uniform rod about different pivots to see that the relevant inertia and centre-of-mass distance both change. State radians and the small-angle approximation before using any elementary period relation. Predict how weaker gravity, greater length, or redistributed mass changes the period before calculating a ratio.
Keep period, frequency, and angular frequency in a three-column table with seconds, hertz, and radians per second. When a result contradicts the qualitative restoring strength, inspect whether a reciprocal was inverted. End by stating the model boundary: undamped or damped, point mass or extended body, small angle or not, and displacement measured from which equilibrium.
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