10Simple Harmonic Motion

📚
SIMPLE HARMONIC MOTION
▢ Periodic Motion:
❖ Definition: Motion in which position repeats after a fixed interval of time
❖ Types:
  • •
    Rectilinear periodic motion
  • •
    Closed curvilinear periodic motion
  • •
    Open curvilinear periodic motion
❖ Examples:

Table 1: Periodic Motion Examples

Motion
Period
Revolution of Earth around Sun
1 year
Rotation of Earth about polar axis
24 hours
▢ Oscillatory / Vibratory Motion:
❖ Definition: To-and-fro or back-and-forth repeated motion about a fixed mean position
❖ Mean Position: Fixed equilibrium position
❖ Extreme Positions: Two fixed limits of oscillation
❖ Relation:
  • •
    Oscillatory motion is constrained periodic motion
  • •
    All oscillatory motions are periodic
  • •
    All periodic motions are not oscillatory
▢ Simple Harmonic Motion:
❖ Definition: Periodic to-and-fro motion about mean position under restoring force/acceleration directed towards mean position and proportional to displacement
❖ Condition:

Table 1: Necessary Condition for SHM

Quantity
Relation
Acceleration
\(a\propto -y\)
Force
\(F\propto -y\)
Restoring force
\(F=-ky\)
Negative sign
Force always directed towards mean position
❖ Important Relation: All SHM are periodic but all periodic motions are not SHM
▢ Types of SHM:

Table 1: Linear vs Angular SHM

Type
Definition
Condition
Differential equation
Linear SHM
Oscillatory motion in straight line about mean position
\(a\propto -y\)
\(\frac{d^2y}{dt^2}+\omega^2y=0\)
Angular SHM
Oscillatory motion about an axis
\(\alpha\propto -\theta\)
\(\frac{d^2\theta}{dt^2}+\omega^2\theta=0\)
❖ Angular Frequency:

Table 1: Frequency Relations

Quantity
Formula
Angular frequency
\(\omega=2\pi f=\frac{2\pi}{T}\)
Frequency
\(f=\frac{1}{T}\)
Time period
Independent of amplitude
▢ Equations of SHM:

Table 1: SHM Formulae

Quantity
Formula
Displacement
\(y=A\sin(\omega t+\phi)\)
Velocity
\(v=A\omega\cos(\omega t+\phi)=\omega\sqrt{A^2-y^2}\)
Acceleration
\(a=-A\omega^2\sin(\omega t+\phi)=-\omega^2y\)
Force
\(F=-mA\omega^2\sin(\omega t+\phi)=-m\omega^2y\)
Momentum
\(p=mA\omega\cos(\omega t+\phi)=m\omega\sqrt{A^2-y^2}\)
Kinetic energy
\(K.E.=\frac{1}{2}mv^2=\frac{1}{2}m\omega^2(A^2-y^2)=\frac{1}{2}k(A^2-y^2)\)
Potential energy
\(P.E.=\frac{1}{2}m\omega^2y^2=\frac{1}{2}ky^2\)
Total energy
\(T.E.=\frac{1}{2}m\omega^2A^2=\frac{1}{2}kA^2\)
❖ At Mean and Extreme Positions:

Table 1: Values at Mean and Extreme Position

Quantity
Mean position
Extreme position
Displacement
\(y=0\)
\(y=A\)
Velocity
\(v=A\omega\) maximum
\(v=0\) minimum
Acceleration
\(a=0\) minimum
\(a=A\omega^2\) maximum
Force
\(F=0\)
\(F=m\omega^2A\) maximum
Kinetic energy
Maximum
0
Potential energy
0
Maximum
Total energy
Constant
Constant
▢ Graphs of SHM:

Table 1: SHM Graphs

Graph
Nature
Displacement-time
Sine curve
Velocity-time
Sine/cosine curve
Acceleration-time
Sine curve
Force-time
Sine curve
Momentum-time
Sine curve
Velocity-displacement
Ellipse
Displacement-acceleration
Straight line with negative slope
Displacement-force
Straight line with negative slope
K.E.-displacement
Parabola
P.E.-displacement
Parabola
Total energy-displacement
Straight line parallel to X-axis
❖ Velocity-Displacement Equation: \(\frac{v^2}{(A\omega)^2}+\frac{y^2}{A^2}=1\)
❖ Acceleration-Displacement Slope: \(\frac{a}{y}=-\omega^2\)
❖ Force-Displacement: \(F=-m\omega^2y\)
▢ Simple Pendulum:
❖ Restoring Force: \(F=-mg\sin\theta\approx -mg\theta=-\frac{mg}{l}y\)
❖ Acceleration: \(a=-\frac{g}{l}y\)
❖ Time Period: \(T=2\pi\sqrt{\frac{l}{g}}\)
❖ Properties:
    ◈ _*type: bullet
  1. Time period independent of mass of bob
  2. Time period independent of shape/material of bob
  3. Time period independent of small amplitude
  4. \(T-l\) graph is parabolic
  5. \(T^2-l\) graph is straight line
❖ General Formula: \(T=2\pi\sqrt{\frac{l}{g*{eff}}}\)
❖ Time Period in Various Cases:

Table 1: Simple Pendulum Special Cases

Case
Time period
Normal condition
\(T=2\pi\sqrt{\frac{l}{g}}\)
Box slides freely down smooth inclined plane at angle \(\theta\)
\(T=2\pi\sqrt{\frac{l}{g\cos\theta}}\)
Lift moving with constant velocity
\(T=2\pi\sqrt{\frac{l}{g}}\)
Lift accelerating upward with acceleration \(a\)
\(T=2\pi\sqrt{\frac{l}{g+a}}\)
Lift accelerating downward with acceleration \(a\)
\(T=2\pi\sqrt{\frac{l}{g-a}}\)
Lift falling freely \(a=g\)
\(T=\infty\); pendulum does not oscillate
Lift accelerating downward with \(a>g\)
\(T=2\pi\sqrt{\frac{l}{a-g}}\)
Length comparable to Earth radius
\(T=2\pi\sqrt{\frac{R}{g}\left(1+\frac{R}{l}\right)}\)
\(l=R\)
\(T=2\pi\sqrt{\frac{R}{2g}}\approx 59.8\ min\)
\(l\to\infty\)
\(T=2\pi\sqrt{\frac{R}{g}}\approx 84.6\ min\)
Bob density \(\sigma\), liquid density \(\rho\)
\(T=2\pi\sqrt{\frac{l}{g(1-\rho/\sigma)}}\)
At height \(h\ll R\) above Earth
\(T\approx \left(1+\frac{h}{R}\right)2\pi\sqrt{\frac{l}{g}}\)
At depth \(d\) inside Earth
\(T=2\pi\sqrt{\frac{l}{g(1-d/R)}}\)
Vehicle accelerating horizontally with \(a\)
\(T=2\pi\sqrt{\frac{l}{\sqrt{a^2+g^2}}}\)
Racing car moving in horizontal circle radius \(R\), speed \(v\)
\(T=2\pi\sqrt{\frac{l}{\sqrt{g^2+v^4/R^2}}}\)
❖ Height and Depth Corrections:

Table 1: Clock Error Formulae

Condition
Change in time period
Loss in time per day
Height \(h\)
\(\frac{\Delta T}{T_0}=\frac{h}{R}\)
\(\frac{86400h}{R}\ sec\)
Depth \(x\)
\(\frac{\Delta T}{T_0}=\frac{x}{2R}\)
\(\frac{86400x}{2R}\ sec\)
Temperature change
\(\frac{\Delta T}{T}=\frac{1}{2}\alpha\Delta\theta\)
Summer: clock slow; winter: clock fast
▢ Spring Oscillation:
❖ Restoring Force: \(F=-ky\)
❖ Acceleration: \(a=-\frac{k}{m}y\)
❖ Time Period: \(T=2\pi\sqrt{\frac{m}{k}}\)
❖ Properties:
  • •
    Time period independent of \(g\)
  • •
    Time period unchanged on Moon / height / depth if same mass-spring system
  • •
    Spring system needs elasticity and inertia
❖ Spring Combination:

Table 1: Series vs Parallel Springs

Feature
Series
Parallel
Force
\(F=F_1=F_2\)
\(F=F_1+F_2\)
Extension
\(y=y_1+y_2\)
\(y=y_1=y_2\)
Equivalent spring constant
\(\frac{1}{k}=\frac{1}{k_1}+\frac{1}{k_2}\)
\(k=k_1+k_2\)
❖ Spring Constant:
◉ _*table:
    ❖ Special Spring Cases:

    Table 1: Spring Oscillation Special Cases

    Case
    Time period
    Same mass attached to one of \(n\) equal parts
    \(T'=\frac{T}{\sqrt n}\)
    Same mass attached to parallel combination of \(n\) parts
    \(T'=\frac{T}{n}\)
    Two masses \(m_1,m_2\) connected to ends of spring constant \(k\)
    \(T=2\pi\sqrt{\frac{m_1m_2}{(m_1+m_2)k}}\)
    Wire length \(L\), area \(A\), Young's modulus \(Y\), mass \(m\)
    \(T=2\pi\sqrt{\frac{mL}{YA}}\)
    ❖ Spring Cut in Ratio:

    Table 1: Spring Cut into m:n Ratio

    Part
    Spring constant
    \(m\) part
    \(k_m=\frac{m+n}{m}k\)
    \(n\) part
    \(k_n=\frac{m+n}{n}k\)
    ▢ Motion Through Earth's Centre:
    ❖ Condition: Body moving inside a tunnel drilled through centre of Earth
    ❖ Restoring Force: \(F=-\frac{GMm}{R^3}y=-\frac{mg}{R}y\)
    ❖ Acceleration: \(a=-\frac{g}{R}y\)
    ❖ _*table:
      ▢ Oscillation of Liquid in U-tube:
      ❖ Symbols:
      • •
        \(L\) = total length of liquid column
      • •
        \(h=L/2\) = height of liquid column
      • •
        \(\rho\) = density of liquid
      • •
        \(A\) = area of cross-section
      ❖ Restoring Force: \(F=-(A\cdot2y)\rho g=-2A\rho gy\)
      ❖ Acceleration: \(a=-\frac{2g}{L}y\)
      ❖ Time Period: \(T=2\pi\sqrt{\frac{L}{2g}}=2\pi\sqrt{\frac{h}{g}}\)
      ❖ Independent Of:
      • •
        Area of cross-section of U-tube
      • •
        Density of liquid
      ❖ Depends On:
      • •
        Length of liquid column
      • •
        Acceleration due to gravity
      ▢ Oscillation of Block in Liquid:
      ❖ Symbols:

      Table 1: Symbols

      Symbol
      Meaning
      \(\sigma\)
      Density of block
      \(\rho\)
      Density of liquid
      \(A\)
      Cross-sectional area of block
      \(M\)
      Mass of block
      \(H\)
      Vertical height of block inside liquid at equilibrium
      ❖ Restoring Force: \(F=-Ay\rho g\)
      ❖ Time Period:

      Table 1: Block in Liquid

      Condition
      Formula
      General
      \(T=2\pi\sqrt{\frac{M}{A\rho g}}\)
      Using block density and height
      \(T=2\pi\sqrt{\frac{\sigma h}{\rho g}}\)
      Using immersed height \(H\)
      \(T=2\pi\sqrt{\frac{H}{g}}\)
      ▢ Time Shortcuts in SHM:

      Table 1: Time Taken in SHM

      Path
      Time
      \(A/2\) from mean position
      \(T/12\)
      \(A/2\) from extreme position
      \(T/6\)
      Extreme to mean
      \(T/4\)
      Mean to extreme
      \(T/4\)
      Complete oscillation displacement
      0
      Complete oscillation distance
      \(4A\)
      ▢ Phase Relations:

      Table 1: Phase Difference in SHM

      Quantities
      Phase difference
      Displacement and velocity
      \(\pi/2\); velocity leads displacement
      Velocity and acceleration
      \(\pi/2\)
      Displacement and acceleration
      \(\pi\)
      Acceleration and force
      0
      K.E. and P.E.
      \(\pi/2\)
      ❖ Initial Phase:

      Table 1: Starting Point and Phase

      Starting point
      \(\phi\)
      Equation
      Mean position
      0°
      \(y=A\sin\omega t\)
      Extreme position
      90°
      \(y=A\cos\omega t\)
      ▢ Energy Frequency:

      Table 1: Energy Oscillation

      Quantity
      Frequency / Period
      SHM frequency
      \(f\)
      K.E. frequency
      \(2f\)
      P.E. frequency
      \(2f\)
      K.E. and P.E. time period
      \(T/2\)
      Total energy frequency
      0
      Total energy
      Same at all positions
      ▢ Undamped and Damped Oscillations:

      Table 1: Oscillation Types

      Type
      Meaning
      Energy / Frequency
      Undamped oscillation
      SHM with constant amplitude
      Total energy constant
      Damped oscillation
      SHM with decreasing amplitude with time
      Frequency decreases; time period increases
      ▢ Read and Digest:

      Table 1: Important SHM Points

      Fact
      Point
      Necessary and sufficient condition
      \(F\propto -y\)
      Displacement direction
      Away from mean position
      Acceleration in SHM
      Changes both magnitude and direction
      If acceleration increases
      Time period remains same
      SHM system requirement
      Elasticity + inertia
      Uniform circular motion
      Periodic but not SHM
      Constants in SHM
      Time period, frequency, angular frequency, total energy, initial phase
      Variables in SHM
      Displacement, velocity, acceleration, force, K.E., P.E.
      Second pendulum
      Time period = 2 s; length = 99.29 cm
      Similarly charged horizontal sheet near charged bob
      Time period increases
      Oppositely charged sheet near charged bob
      Time period decreases
      Spring time period in accelerating vehicle
      Same as stationary vehicle
      Simple pendulum in accelerating vehicle
      Time period decreases
      Girl on swing stands up
      Time period decreases
      Friend sits beside girl on swing
      Time period remains same
      Change in amplitude
      Does not change time period of simple pendulum
      Hollow sphere filled with water, water flows out
      Period first increases then decreases
      Hollow sphere pendulum with mercury, little mercury drained
      Time period increases
      Simple pendulum on Moon
      Frequency becomes \(\frac{f}{\sqrt6}\)
      Length-time period graph
      Parabola
      \(T^2-l\) graph
      Straight line
      ▢ High-Yield Recall:

      Table 1: SHM One-Liners

      Fact
      Answer
      Periodic motion
      Position repeats after fixed time
      Oscillatory motion
      To-and-fro motion about mean position
      SHM condition
      \(F\propto -y\)
      Restoring force
      \(F=-ky\)
      Linear SHM equation
      \(\frac{d^2y}{dt^2}+\omega^2y=0\)
      Angular SHM equation
      \(\frac{d^2\theta}{dt^2}+\omega^2\theta=0\)
      Angular frequency
      \(\omega=2\pi f=\frac{2\pi}{T}\)
      Displacement
      \(y=A\sin(\omega t+\phi)\)
      Velocity
      \(v=\omega\sqrt{A^2-y^2}\)
      Acceleration
      \(a=-\omega^2y\)
      K.E.
      \(\frac{1}{2}m\omega^2(A^2-y^2)\)
      P.E.
      \(\frac{1}{2}m\omega^2y^2\)
      Total energy
      \(\frac{1}{2}m\omega^2A^2\)
      At mean position
      Velocity maximum; acceleration zero
      At extreme position
      Velocity zero; acceleration maximum
      Velocity-displacement graph
      Ellipse
      Acceleration-displacement graph
      Straight line
      Simple pendulum time period
      \(T=2\pi\sqrt{\frac{l}{g}}\)
      Spring time period
      \(T=2\pi\sqrt{\frac{m}{k}}\)
      U-tube liquid time period
      \(T=2\pi\sqrt{\frac{L}{2g}}\)
      Earth tunnel time period
      84.6 min
      Earth tunnel centre time
      21 min
      Second pendulum length
      99.29 cm
      K.E. frequency
      \(2f\)
      Total energy frequency
      0
      Undamped oscillation
      Constant amplitude
      Damped oscillation
      Decreasing amplitude
      Q1.
      Simple pendulum has amplitude R and period T. Time to move from y = R to y = R/2 is
      📅CEE MBBS 2025
      Q2.
      The SHM of a particle is given by an equation y = 4 sinωt + 3 cosωt. The amplitude is:
      📅CEE Nursing 2024
      Q3.
      The phase difference between velocity and acceleration in SHM is:
      📅CEE BDS 2023
      Q4.
      In SHM, the period of oscillation of mass M suspended from a spring is 1 second, then time period of the oscillation of mass 4M will be:
      📅CEE BDS 2021
      Q5.
      The graph plotted between acceleration and displacement in SHM will be a:
      📅BP 2011
      Q6.
      Two bodies execute SHM simultaneously. Their resultant amplitude is maximum when their phase difference is:
      📅BP 2009
      Q7.
      The time period of a simple pendulum is:
      📅BP 2009
      Q8.
      The SHM of a particle is \(y=3\sin\omega t+4\cos\omega t\). Its amplitude is:
      📅MOE 2014
      Q9.
      The kinetic energy of a particle executing SHM when its displacement is one-third of its amplitude is:
      📅MOE 2012
      Q10.
      If the kinetic energy of a body executing SHM is one-third of its potential energy, the displacement is approximately what percentage of the amplitude?
      📅MOE 2009
      Q11.
      For a simple pendulum of length \(l\) and period \(T\), the acceleration due to gravity \(g\) is:
      📅I.E. 2012
      Q12.
      For a body executing SHM, at the mean position:
      📅I.E. 2012
      Q13.
      For a particle executing SHM, \(K=K_0\cos^2\omega t\). The maximum potential energy is:
      📅I.E. 2013
      Q14.
      A simple pendulum has period \(T_1\) at Earth’s surface and period \(T_2\) at height \(R\) above the surface, where \(R\) is Earth’s radius. Find \(T_2/T_1\).
      📅IOM 2014
      Q15.
      The time period of a simple pendulum doubles when:
      📅IOM 2012
      Q16.
      A pendulum eventually stops oscillating because of:
      📅IOM 2011
      Q17.
      In SHM, the particle’s speed is minimum at the:
      📅KU 2014
      Q18.
      A hollow sphere containing water is suspended by a thread and made to oscillate. As water slowly flows out of a small hole at the bottom, the period initially:
      📅KU 2014
      Q19.
      A pendulum clock taken to high altitude will keep correct time if:
      📅KU 2014
      Q20.
      If a mass \(M\) oscillating on a spring has period 1 s, the period for a mass \(4M\) on the same spring is:
      📅KU 2012
      Q21.
      The differential equation of SHM is \(\frac{d^2y}{dt^2}+100y=0\). Its frequency is:
      Q22.
      The period of a simple pendulum doubles when:
      📅IOM 2012
      Q23.
      A particle in SHM has maximum acceleration \(a_0\) and maximum speed \(v_0\). Its amplitude is:
      Q24.
      A pendulum hangs from the ceiling of a train at rest and has period \(T\). When the train accelerates horizontally with constant acceleration \(a\), its period:
      Q25.
      A planet has twice Earth’s mass and twice Earth’s diameter. A pendulum that has period 1 s on Earth has period on this planet:
      Q26.
      A seconds pendulum has a period of 2 s with a 50 g hollow bob. The hollow bob is replaced by a 100 g solid bob of the same radius. The period becomes:
      Q27.
      A 1 kg weight extends a light spring by 9.8 cm. The period of oscillation is:
      Q28.
      A particle performs SHM with amplitude \(a\). Its displacement when kinetic and potential energies are equal is:
      Q29.
      A simple pendulum with bob mass \(m\) swings with angular amplitude \(40^\circ\). When its angular displacement is \(20^\circ\), the tension in the string is:
      📅IOM 2008
      Q30.
      The maximum speed and maximum acceleration of a particle in SHM are \(4\ m/s\) and \(2\ m/s^2\). Its period is:
      Q31.
      A body executes SHM of amplitude \(A_0\). At what displacement from the mean position is its kinetic energy half its maximum value?
      Q32.
      A body executes SHM of amplitude \(A_0\). At what displacement from the mean position is its speed half the maximum speed?
      Q33.
      A spring of force constant \(k\) supports mass \(m\) and oscillates at frequency \(n\). The spring is cut into two equal halves; the same mass is suspended from one half. New frequency:
      Q34.
      Two blocks of masses \(m_1=2\ kg\) and \(m_2=8\ kg\) are connected by a spring of constant \(k=1000\ N/m\) on a smooth surface. Find the frequency of their relative oscillations.
      📅BPKIHS
      Q35.
      A spring of length \(l\) has spring constant \(k\). It is cut into two pieces of lengths \(l_1\) and \(l_2\), with \(l_1=nl_2\). The spring constant of the first piece is:
      Q36.
      Starting at the mean position, a particle in SHM takes \(t_1\) to cover the first half of its amplitude and \(t_2\) to cover the remaining half to an extreme position. Then:
      Q37.
      A stone falls through a frictionless tunnel along Earth’s diameter. Approximately how long does it take to travel from the surface to a depth of \(R/2\)?
      Q38.
      A spring has energy \(U_0\) when extended by 2 cm. Its stored energy at 10 cm extension is:
      Q39.
      A simple pendulum of length \(L\) is released from a small angular displacement \(a\) at \(t=0\). Its angular displacement at time \(t\) is:
      📅BPKIHS
      Q40.
      Equal masses oscillate vertically on separate springs with spring constants \(k_1\) and \(k_2\). If their maximum speeds are equal, the amplitude ratio \(A_M/A_N\) is:
      📅I.E.•BPKIHS
      Q41.
      A simple pendulum has a brass bob with period \(T\). It is immersed in a non-viscous liquid whose density is one-tenth the density of brass. Its new period is:
      Q42.
      The kinetic energy of a block oscillating on a spring is independent of:
      Q43.
      A simple pendulum’s period increases by 0.1% when taken to the peak of a mountain. If Earth’s radius is 6400 km, the peak’s height is approximately:
      Q44.
      A simple pendulum’s period increases by 0.01% when taken into a mine. For Earth’s radius 6400 km, estimate the depth of the mine:
      Q45.
      A body executes SHM. Its speeds at displacements 4 cm and 5 cm are 10 cm/s and 8 cm/s respectively. Find its period:
      Q46.
      A simple pendulum has a solid steel-sphere bob. If it is replaced by a hollow steel-sphere bob of the same mass and material, its period (with the suspension length unchanged) will:
      Q47.
      During SHM, when a particle lies midway between the mean and an extreme position, the ratio of kinetic to potential energy is:
      Q48.
      A particle of mass 10 g executes SHM with amplitude 0.5 m and period \(\pi/5\ s\). The maximum force acting on it is:
      Q49.
      A 5 kg mass hangs from a spring and oscillates with time period \(2\pi\ s\). What equilibrium extension does this mass produce in the spring?
      Q50.
      A simple harmonic oscillator has displacement 0.02 m and acceleration 2 m/s² at a particular instant. Its angular frequency is:
      Q51.
      Two springs oscillate separately with periods 3 s and \(x\) s. Connected in series, their period is 5 s; connected in parallel, the period is \(y\) s. Find \((x,y)\).
      Q52.
      A uniform spring of force constant \(k\) is cut into two parts in the length ratio 1:2. What is the spring constant of the longer part?
      Q53.
      A pendulum is raised to an angular displacement of \(4^\circ\) and released from rest. Taking cosine as the reference, its initial phase is:
      Q54.
      If \(T\) is the period of SHM, which relation is true?
      Q55.
      Two pendulums of lengths 1 m and 1.21 m start swinging together. How many oscillations will the longer pendulum complete before they next swing together in phase?
      Q56.
      If the length of a pendulum increases by 21%, its time period increases by:
      Q57.
      Two simple harmonic motions are \(x_1=a\sin(\omega t)\) and \(x_2=b\cos(\omega t)\). Their phase difference is:
      📅MOE 2009
      Q58.
      A body of mass 4m kg is suspended from a spring and has period \(T_1\). If replaced by a body of mass m kg, the period is \(T_2\). Find \(T_1/T_2\).
      📅KU 2010
      Q59.
      The kinetic energy of a particle executing SHM is maximum at displacement equal to:
      📅MOE 2066
      Q60.
      The period of a simple pendulum inside a satellite orbiting Earth is:
      📅MOE 2063
      Q61.
      A tunnel passes through Earth’s centre. A stone dropped into it reaches the centre. At the centre, its:
      📅MOE 2062
      Q62.
      For SHM of amplitude \(A\), kinetic and potential energies are equal at displacement:
      📅MOE 2010
      Q63.
      A frictionless hole is bored along Earth’s diameter. When a ball is dropped into one end, it:
      📅MOE 2061
      Q64.
      A 3 kg block on a spring executes SHM described by \(x=2\cos(50t)\) in SI units. Find the spring constant:
      📅I.E. 2006
      Q65.
      If a simple pendulum’s length increases by 44%, its period increases by:
      📅I.E. 2004
      Q66.
      A particle in SHM has maximum speed \(v_{\max}\). Its average speed over one complete oscillation is:
      📅BPKIHS 2009
      Q67.
      For a simple pendulum, the graph between \(T^2\) and length \(l\) is:
      📅BPKIHS 2004
      Q68.
      The motion of a particle in SHM is \(x=0.01\sin(100\pi(t+0.005))\) (SI units). Its period is:
      📅BPKIHS 1996
      Q69.
      At which position during SHM is kinetic energy maximum?
      📅KU 2016
      Q70.
      In SHM, kinetic energy is minimum at the:
      📅KU 2017