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EAPCET Physics · Intermediate first year

Oscillations

Oscillations covers simple harmonic motion, the spring–mass system and the simple pendulum, and energy in oscillating systems.

Subject
Physics
Taught in
Intermediate first year
Physics in the paper
40 of 160 questions

Key formulas and results

  • SHM: a=−ω2xa = -\omega^{2}x
  • x=Asin⁡(ωt+ϕ)x = A\sin(\omega t + \phi) and v=ωA2−x2v = \omega\sqrt{A^{2} - x^{2}}
  • Spring–mass: T=2πm/kT = 2\pi\sqrt{m/k}; simple pendulum: T=2πl/gT = 2\pi\sqrt{l/g}
  • Total energy =12kA2= \tfrac{1}{2}kA^{2}, constant throughout the motion
  • Springs in series: 1k=1k1+1k2\frac{1}{k} = \frac{1}{k_{1}} + \frac{1}{k_{2}}; in parallel: k=k1+k2k = k_{1} + k_{2}
  • Velocity and acceleration in SHM: v=Aωcos⁡(ωt+ϕ)v = A\omega\cos(\omega t + \phi), a=−Aω2sin⁡(ωt+ϕ)a = -A\omega^{2}\sin(\omega t + \phi)
  • Kinetic energy at displacement xx: KE=12mω2(A2−x2)\text{KE} = \tfrac{1}{2}m\omega^{2}(A^{2} - x^{2}); potential energy PE=12mω2x2\text{PE} = \tfrac{1}{2}m\omega^{2}x^{2}; total energy 12mω2A2\tfrac{1}{2}m\omega^{2}A^{2}
  • Time to sweep a phase angle Δϕ\Delta\phi: t=Δϕω=Δϕ2πTt = \frac{\Delta\phi}{\omega} = \frac{\Delta\phi}{2\pi}T

What this chapter covers

  • Periodic and simple harmonic motion
  • Displacement, velocity and acceleration in SHM
  • Spring–mass system
  • Simple pendulum
  • Energy in SHM

How it is usually asked

  • Find a time period
  • Kinetic and potential energy at a position
  • Phase and displacement
  • Combinations of springs

Understanding Oscillations

Simple harmonic motion happens whenever the restoring force is proportional to the displacement and directed towards the mean position, F = −kx. This gives a = −ω²x with ω = √(k/m), and the displacement is a sine function of time. The period depends only on the system (mass and spring constant, or length and g), not on the amplitude.

Speed is greatest at the mean position and zero at the extremes, while acceleration is the opposite: zero at the mean position and greatest at the extremes. Energy keeps swapping between kinetic and potential forms, but the total stays ½kA² (equivalently ½mω²A²). Springs combine opposite to resistors: in parallel their constants add, in series their reciprocals add.

EAPCET questions usually ask for a period, a speed or energy at a given displacement, the effect of changing mass, length or g, or the spring constant of a combination. Many reduce to scaling: T ∝ √m for a spring and T ∝ √(l/g) for a pendulum, so quadrupling the mass or the length doubles the period, and a pendulum taken to the moon, where g is about one-sixth as large, swings more slowly.

Worked example

EAPCET-style question

A particle executes SHM with amplitude 0.04 m and angular frequency 5 rad/s. Its speed when the displacement is 0.02 m is

  1. (1)0.13 m/s0.1\sqrt{3}\ \text{m/s}
  2. (2)0.2 m/s
  3. (3)0.1 m/s
  4. (4)0.053 m/s0.05\sqrt{3}\ \text{m/s}

Solution

  1. Use v=ωA2−x2v = \omega\sqrt{A^{2} - x^{2}}.
  2. A2−x2=0.0016−0.0004=0.0012 m2A^{2} - x^{2} = 0.0016 - 0.0004 = 0.0012\ \text{m}^{2}.
  3. 0.0012=0.023 m\sqrt{0.0012} = 0.02\sqrt{3}\ \text{m}.
  4. v=5×0.023=0.13 m/s≈0.17 m/sv = 5 \times 0.02\sqrt{3} = 0.1\sqrt{3}\ \text{m/s} \approx 0.17\ \text{m/s}.

Answer: 0.13 m/s0.1\sqrt{3}\ \text{m/s}

Shortcuts and results to remember

  • From the mean position to x=A/2x = A/2 takes T/12T/12, from A/2A/2 to the extreme takes T/6T/6, and mean to extreme takes T/4T/4.
  • x=a+bsin⁡2ωtx = a + b\sin^{2}\omega t is SHM about x=a+b/2x = a + b/2 with amplitude b/2b/2 and angular frequency 2ω2\omega.
  • The phase difference between two particles doing SHM with the same ω\omega never changes with time.
  • Find the initial phase from the starting position and direction of motion using the reference circle: equal phase angles are swept in equal times.

More practice questions

Practice question 1

A particle moves as x=Acos⁡(ωt+π/3)x = A\cos(\omega t + \pi/3). The minimum time it takes to reach the left extreme position (period TT) is

  1. (1)T/6T/6
  2. (2)T/4T/4
  3. (3)T/3T/3
  4. (4)T/2T/2

Solution

  1. At t=0t = 0, x=Acos⁡(π/3)=A/2x = A\cos(\pi/3) = A/2 and v=−Aωsin⁡(π/3)<0v = -A\omega\sin(\pi/3) < 0, so it moves left
  2. The left extreme x=−Ax = -A needs ωt+π/3=π\omega t + \pi/3 = \pi
  3. ωt=2π/3\omega t = 2\pi/3, so t=2π/32π/T=T/3t = \dfrac{2\pi/3}{2\pi/T} = T/3

Answer: T/3T/3

Practice question 2

The displacement of a particle is x=3+4sin⁡2ωtx = 3 + 4\sin^{2}\omega t. Its mean position and amplitude are

  1. (1)x=3x = 3, amplitude 4
  2. (2)x=5x = 5, amplitude 2
  3. (3)x=3x = 3, amplitude 2
  4. (4)x=7x = 7, amplitude 4

Solution

  1. Use sin⁡2ωt=1−cos⁡2ωt2\sin^{2}\omega t = \frac{1 - \cos 2\omega t}{2}
  2. x=3+2(1−cos⁡2ωt)=5−2cos⁡2ωtx = 3 + 2(1 - \cos 2\omega t) = 5 - 2\cos 2\omega t
  3. This is SHM about x=5x = 5 with amplitude 2 and angular frequency 2ω2\omega

Answer: x=5x = 5, amplitude 2

Common mistakes to avoid

  • Using frequency f where angular frequency is needed — ω = 2πf, so check which one the question gives.
  • Combining springs like resistors — springs in series follow 1/k = 1/k₁ + 1/k₂ and in parallel k = k₁ + k₂.
  • Thinking a pendulum’s period depends on the mass of the bob — it depends only on l and g for small oscillations.
  • Placing maximum acceleration at the mean position — acceleration is maximum (Aω²) at the extremes and zero at the mean position.

How to prepare it

Maximum speed is Aω and maximum acceleration is Aω² — many questions need nothing more.

Once the formulas feel familiar, test them the way the exam does: practise Oscillations questions chapter-wise, then attempt a full previous year paper to see whether it holds up under a 3-hour timer.

Related Physics chapters

See every chapter in the EAPCET Physics syllabus.

Practise Oscillations chapter-wise

Chapter-wise quizzes show exactly which topics in this chapter you still get wrong — before a full paper does.

Frequently asked questions

Is Oscillations part of the EAPCET Physics syllabus?
Yes. Oscillations is taught in Intermediate first year Physics, and the EAPCET Engineering syllabus is based on the Intermediate first- and second-year syllabus. Check the official notification each year for any changes.
How should I prepare Oscillations for EAPCET?
Maximum speed is Aω and maximum acceleration is Aω² — many questions need nothing more.
How many Physics questions are there in EAPCET?
40 of the 160 questions in the Engineering paper are Physics, spread across 30 chapters.
What mistakes do students make in Oscillations?
Using frequency f where angular frequency is needed — ω = 2πf, so check which one the question gives. Combining springs like resistors — springs in series follow 1/k = 1/k₁ + 1/k₂ and in parallel k = k₁ + k₂. Thinking a pendulum’s period depends on the mass of the bob — it depends only on l and g for small oscillations. Placing maximum acceleration at the mean position — acceleration is maximum (Aω²) at the extremes and zero at the mean position.