Period vs Mass
T ∝ √m — square-root curve.
Key Notes
For spring-mass SHM: T = 2π·√(m/k). Period proportional to √m.
Doubling mass increases T by √2 (~1.41×).
T² vs m gives a straight line through origin — slope = 4π²/k. Lab method to measure k.
For a SIMPLE PENDULUM: T does NOT depend on mass — period depends only on L and g.
Why difference? Spring force scales with displacement (not mass), so heavier mass takes longer to accelerate. Pendulum gravity scales with mass (= weight), so mass cancels out.
In rotational systems, 'mass' is replaced by moment of inertia I.
Same principle: increasing inertia at constant restoring force/torque ⇒ slower oscillation.
Useful for designing tuning forks, accelerometers, mechanical clocks.
Formulas
Spring-mass period
Square-root dependence on mass.
Linearised form
Plot T² vs m — slope = 4π²/k.
Mass independence (pendulum)
Simple pendulum period doesn't depend on bob mass.
Important Points
Spring-mass: T ∝ √m. Doubling mass changes T by √2.
Pendulum: T independent of m (gravity force scales with m, cancels acceleration).
Why the difference: in spring, F = −kx (independent of m). In gravity, F = mg (dependent on m). Newton's 2nd law: a = F/m. For pendulum, m cancels; for spring, it doesn't.
Slope of T² vs m graph gives k: very useful in introductory physics labs.
Real spring has its own mass m_s; effective m = m + m_s/3 (Rayleigh's correction).
Common pitfall: thinking pendulum period changes with mass. It doesn't (in ideal case).
Period vs Mass notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 11 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.