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.
      A fan makes 10 revolutions in 3 second which is just switched on. Considering uniform acceleration the number of revolution made by fan in next 3 second is:
      📅BP 2010
      Q2.
      The spokes are used in bicycle wheel to [BP 201 1]
      📅BP 201 1
      Q3.
      A small mass of 10 gm, lies in a hemispherical bowl of radius 0.4 m at a height of 0.2 m from the bottom of the bowl. The mass will be in equilibrium of the bowl rotates at an angular speed of
      📅BP 2009
      Q4.
      A thin uniform rod of mass 'm' moves translationally with acceleration 'a' due to two antiparallel force of lever arm '. One force is of magnitude F and acts at one extreme end. The length of the rod is [BP 2009]
      📅BP 2009
      Q5.
      A wire of length / and mass 'm' is bent in the form of a rectangle ABCD with 2. The moment of inertia of this wire frame about the side BC is :
      📅BP 2009
      Q6.
      A billiard ball is hit by a cue at a height " above the center. It acquires a linear velocity Vo. Mass of the ball is m and radius is r. The angular velocity acquired by the ball is: [BP 2009)
      📅BP 2009)
      Q7.
      A ring, a dice, solid sphere, hollow sphere are dropped from the same inclined plane of same height then which one of the following reaches the ground first [MOE 2014)
      📅MOE 2014)
      Q8.
      The moment of inertia of a body of mass M about a given axis is I. What is the radius of gyration? [MOE 2014)
      📅MOE 2014)
      Q9.
      The torque due to gravitational force on body about its centre of mass is: [MOE 2014)
      📅MOE 2014)
      Q10.
      Two forces of 2N and 4N attached at the ends of a 0.5 meter rod act vertically downwards. A third force will keep the system in equilibrium if applied at a point between the ends of the ro magnitude, direction and position of the third force will be: [MOE 2011]
      📅MOE 2011
      Q11.
      11. Two point masses of 1 kg and 2 separated by 0.5 m constitute a system The distance of the centre of mass of the system from 1 kg mass is: [IMOE 20131
      📅IMOE 20131
      Q12.
      12. A circular body of mass 2 kg of radius I then of inertia about diameter is? [MOE 2011
      📅MOE 2011
      Q13.
      13. Moment of inertia doesn't depend upon [MOE 2010
      📅MOE 2010
      Q14.
      14. If 'M' and 'r' are respectively the mass of electrons and radius of the orbit in which the electron revolves about the nucleus, the moment of inertia of electron will be: [MOE 2009]
      📅MOE 2009
      Q15.
      15. When a body rolls downs an inclined plane. The total potential energy of the body changes into:
      📅IE 2011
      Q16.
      16. If no internal force is applied in a body the velocity of the centre of mass: [IOM 2013]
      📅IOM 2013
      Q17.
      17. The product of moment of inertia and [IOM 2013] angular acceleration gives,
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      Q18.
      18, A cylinder has mass "M" a length T and Radius 'R' then M.I. about own axis is:
      📅IOM 2012
      Q19.
      Two bodies of masses m, and m; move in circles of radii r, and ra respectively. If they complete the circles in equal time, the ratio of their angular speed @ [KU 2014]
      📅KU 2014
      Q20.
      . A uniform heavy disc is rotating with a constant angular velocity about a vertical axis through its center. Some wax is dropped gently on the disc near to the edge. The angular velocity of the disc [KU 2012]
      📅KU 2012
      Q21.
      A uniform metal disc of radius R lies in XY - plane and rotates with uniform angular velocity w about the Z - axis, the total induced EMF between the center and the rim of the disc is equal to; [KU 2011]
      📅KU 2011
      Q22.
      Two masses of 1 kg and 2 kg are 9 m apart and make tw mass from 1 kg mass will be at [Bangladesh 09]
      📅Bangladesh 09
      Q23.
      3. A uniform disc is rotating at a constant speed about a vertical axis through its centre. Some wax is gently dropped on the disc, the angular velocity of the disc[KU 091
      📅KU 091
      Q24.
      . A circular disc of mass m and radius r is rotating about its axis with uniform speed of v. What is its kinetic energy? [TOM 04]
      📅TOM 04
      Q25.
      When the size of the earth is reduced to half, mass remaining same, the time period of the earth rotation will be: [IOM 031
      📅IOM 031
      Q26.
      A rotating disc has ...., kinetic energy, i mass is M & velocity is V [IOM 98
      📅IOM 98
      Q27.
      A fly-wheel of mass 10 kg and radius 50 cm is rotating with constant angular speed of @ with its kinetic energy 20 Joule. The angular speed of flywheel is [MOE 066]
      📅MOE 066
      Q28.
      The body applied with constant torque changes the angular momentum Io to final angular momentum 41, in 3 sec. then find torque [MOE 2008]
      📅MOE 2008
      Q29.
      Kinetic energy of a body is given by 1/2 mv. Which one of the following expression is correct for the kinetic energy of the rigid body where I andw represent the moment of intertia and angular velocity of the rigid body? [MOE 2065]
      📅MOE 2065
      Q30.
      If a body starts from rest with angular acceleration a= 6t. What is time taken to complete 10 revolution?
      Q31.
      If there is a change of angular momentum from 2 J to 4 J in 4 sec. Then the torque is [TE-04)
      📅TE-04)
      Q32.
      When torque acting upon a system is zer Which of the following will be constant? [TE-051
      📅TE-051
      Q33.
      A shell at rest explodes. The centre of mass of the fragments
      📅IE-08•BP 2017
      Q34.
      The moment of inertia of a disc of mass M and radius R about an axis which is tangent to the circumference of the disc and parallel to its diameter is:
      📅BPKIHS-08
      Q35.
      A particle of mass m and radius of gyration k is rotating with an angular acceleration o. The torque acting on it is
      Q36.
      The centre of gravity of a body [BPKTHS-94)
      📅BPKTHS-94)
      Q37.
      Radius of Gyration of an uniform rod about an axis through its middle is [BPKIHS-94]
      📅BPKIHS-94]
      Q38.
      Let I, and I be the moments of inertia of two bodies of identical geometrical shape, the first made of almunium and the second of iron [BPKIHS-95]
      📅BPKIHS-95
      Q39.
      Three point masses each of mass m are placed at the corners of an equilateral triangle of side /. The moment of inertia of system about an axis along one side of the triangle is [BPKIHS-96]
      📅BPKIHS-96
      Q40.
      Ratio of the angular velocity of the earth about its axis and the hour hand of a clock is [BPKIHS 1999]
      📅BPKIHS 1999
      Q41.
      If the radius of the earth's orbit is made one fourth, the duration of year will become [BPKIHS 2000]
      📅BPKIHS 2000
      Q42.
      The moment of inertia of a circular ring of mass M and radius R about its diameter is
      Q43.
      The moment of inertia of a thin rod of mass M, length L, about an axis passing through a point from one end and perpendicular to length is
      Q44.
      The moment of inertia of a solid sphere of mass M radius R about its diameter is
      Q45.
      The M.I of a solid cylinder of length /, radius R about its geometrical axis is same as about equatorial axis, then the ratio of R and I will be
      Q46.
      A uniform metallic disc of moment of inertia Io about its own axis is melted and a uniform ring of equal radius is then casted from it. Then, M.I of the ring about its diameter will be
      Q47.
      A uniform metallic disc has its M.I I. about its diameter. Then its M.I about an axis through its rim perpendicular to the plane will be
      Q48.
      The radii of two steel balls are R and 2R. Then, their moment of inertia about their diameters are in the ratio [KU 2009]
      📅KU 2009
      Q49.
      A circular portion of diameter R is cut out from the edge of a uniform disc of mass M and radius R. The M.I of the remaining portion of the disc about an axis passing through the centre O of the disc and perpendicular to its plane is
      Q50.
      A uniform rod of mass M and length L is rotating with angular speed ωo with two beads of mass m on either side of the axis passing through its centre and perpendicular to its length. The beads slide outward as it rotates. What will be the final angular speed when the beads reach the ends ?
      Q51.
      Three thin rods each of length L and mass M are placed along X, Y and Z-axis in such a way that one end of rod is at the origin. The moment of inertia of the system about Z-axis is
      Q52.
      The M.I of two spheres of equal masses about their respective diameters are same. If one of them is solid and other is hollow, then the ratio of their radii (solid to hollow), will be
      Q53.
      Two circular discs of same mass an thickness are made from metals having densities d, and dy respectively. The ratio of their moments of inertia about the central axis will be
      Q54.
      A wheel of moment of inertia 5x10 kg -m is making 20 rev/sec. The torque required to stop it in 10 sec is
      Q55.
      A thin hollow cylinder open at both ends, Slides without rotating rolls without slipping with the same ii. speed. The ratio of K.E in the two cases is:
      Q56.
      A solid sphere of mass M is rolling on a horizontal surface without sliding with velocity v. Its kinetic energy will be
      Q57.
      A solid sphere of mass M is rotating about its diameter and linear velocity of a point on its equator is v. Then its kinetic energy will be
      Q58.
      A body rolling without sliding has its rotational kinetic energy equal to 40% of total energy. Then body should be
      Q59.
      A solid spherical ball rolls on a table. Ratio of rotational. K.E to the total K.E is
      Q60.
      The least coefficient of friction for an inclined plane of inclination a with the horizontal in order that a solid cylinder will roll down without slipping is
      Q61.
      A wheel of mass 10kg has a moment of inertia 160kg-m' about its own axis. The radius of gyration is:
      Q62.
      The radius of gyration of a solid disc of mass 1kg and radius 50cm about an axis through centre of mass and perpendicular to its face is
      Q63.
      A uniform circular disc, 20g is rotating about its own vertical axis at 30 rpm. When 20g sand falls on its surface at distance 5cm from the centre of the disc, the rate of rotation decreases to 24 rpm. Then the radius of the disc should be:
      Q64.
      A particle performs uniform circular motion with an angular moment L. If the frequency of particle's motion is doubled and its kinetic energy is halved, the angular momentum becomes:
      Q65.
      A constant torque acting on a uniform circular wheel changes its . angular omentum from Jo to 43, in 4 seconds. The magnitude of the torque is:
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      Q66.
      A flywheel of moment of inertia 0.5kgm i rotating 300 rpm initially comes to rest in 10 seconds under constant retarding torque. Then the number of revolutions made by the wheel until rest is:
      Q67.
      A sphere of mass 2kg and radius 5cm is rotating at the rate of 300rpm. Then th torque required to stop it in 2nt revolutions is:
      Q68.
      Two particles A and B initially at rest move towards each other under a mutual force of attraction. At the instant when velocity of A is v and that of B is 2v, the velocity of centre of mass of the system is:
      Q69.
      Two particles of masses m, and my are at distance x. Then, their centre of mass lies at distance from my.
      Q70.
      Out of two particles of masses m, and m₂, the towards their centre of mass. What is the displacement of centre of mass?
      Q71.
      Two blocks of masses 5kg and 2kg ar placed on a frictionless surface and connected by a spring. An external kick gives a velocity of 14 m/s to the heavier block in the direction of lighter one Calculate the velocity gained by the centre of mass.
      Q72.
      A shell is fixed a gun with a muzzle velocity u m/s at an angle 0 with the horizontal. At the top of the trajectory, the shell explodes into two fragments P and Q of equal mass. If the speed of fragment P immediately after explosion becomes zero where does the fragment Q hit the ground from the point of projection?
      Q73.
      . A circular plate of uniform thickness has diameter of 56cm. A circular portion of diameter 42cm is removed from one edge as shown in the fig. The centre of mass of remaining from the centre of plate will be
      Q74.
      83. Two masses of 1kg and 2kg are 9m apart and make a two body system. Their centre of mass from 1kg mass will be at [MOE]
      📅MOE
      Q75.
      Let F be a force acting on a particle having position vector 'r' . Let 't' be the torque of this force about the origin, then [KU 2015]
      📅KU 2015
      Q76.
      85. If a gymnast on a rotating stool with his arms outstretched suddenly lower his arms [TOM 2015]
      📅TOM 2015