9Rotational motion

📚
ROTATIONAL MOTION
▢ Translational vs Rotational Analogy:

Table 1: Linear and Angular Quantities

Translational motion
Rotational motion
Relation
Linear displacement \(s\)
Angular displacement \(\theta\)
\(\vec s=\vec\theta\times\vec r\)
Linear velocity \(v=\frac{ds}{dt}\)
Angular velocity \(\omega=\frac{d\theta}{dt}\)
\(\vec v=\vec\omega\times\vec r\)
Linear acceleration \(a=\frac{dv}{dt}\)
Angular acceleration \(\alpha=\frac{d\omega}{dt}\)
\(\vec a=\vec\alpha\times\vec r\)
Mass \(m\)
Moment of inertia \(I\)
\(I=mr^2\)
Linear momentum \(p=mv\)
Angular momentum \(L=I\omega\)
\(\vec L=\vec r\times\vec p\)
Force \(F=\frac{dp}{dt}=ma\)
Torque \(\tau=\frac{dL}{dt}=I\alpha\)
\(\vec\tau=\vec r\times\vec F\)
Work \(W=\int F\,ds\)
Work \(W=\int \tau\,d\theta\)
—
Power \(P=Fv\)
Power \(P=\tau\omega\)
—
K.E. \(=\frac{1}{2}mv^2\)
K.E. \(=\frac{1}{2}I\omega^2\)
—
▢ Rotational Kinematics:

Table 1: Linear vs Angular Equations

Linear equation
Angular equation
\(v=u+at\)
\(\omega=\omega_0+\alpha t\)
\(v^2=u^2+2as\)
\(\omega^2=\omega_0^2+2\alpha\theta\)
\(s=ut+\frac{1}{2}at^2\)
\(\theta=\omega_0t+\frac{1}{2}\alpha t^2\)
\(s_n=u+\frac{a}{2}(2n-1)\)
\(\theta_n=\omega_0+\frac{\alpha}{2}(2n-1)\)
❖ Constant Angular Acceleration from Rest:

Table 1: Rotational Ratios

Quantity
Ratio
Angle / revolutions up to successive equal time intervals
\(1^2:2^2:3^2:...:n^2\)
Angle / revolutions in successive equal time intervals
\(1:3:5:...:(2n-1)\)
Angular velocities at ends of successive equal time intervals
\(1:2:3:...:n\)
Time to complete successive equal revolutions from start
\(\sqrt1:\sqrt2:\sqrt3:...:\sqrt n\)
Time for successive equal revolutions
\((\sqrt1-\sqrt0):(\sqrt2-\sqrt1):(\sqrt3-\sqrt2):...:(\sqrt n-\sqrt{n-1})\)
▢ Moment of Inertia:
❖ Definition: Rotational analogue of mass; depends on mass and distribution of mass about axis of rotation
❖ Theorems:
◉ _*table:
    ❖ Radius of Gyration:
    ◉ Formula: \(I=MK^2\)
    ◉ Ratio: \(n=\frac{K^2}{R^2}\)
    ◉ Note: Radius of gyration is independent of mass
    ❖ Moment of Inertia Table:

    Table 1: Important M.I. Formulae

    Body
    Axis
    Moment of inertia
    \(n=K^2/R^2\)
    Uniform rod
    Through C.G. and perpendicular to length
    \(\frac{ML^2}{12}\)
    \(\frac{1}{12}\)
    Uniform rod
    Through end and perpendicular to length
    \(\frac{ML^2}{3}\)
    \(\frac{1}{3}\)
    Rectangular lamina
    Through C.G. and perpendicular to plane
    \(\frac{M(l^2+b^2)}{12}\)
    \(\frac{1}{12}\)
    Ring / circular loop
    Through C.G. and perpendicular to plane
    \(MR^2\)
    1
    Ring / circular loop
    Axis perpendicular to plane through edge
    \(2MR^2\)
    2
    Ring / circular loop
    About diameter
    \(\frac{MR^2}{2}\)
    \(\frac{1}{2}\)
    Ring / circular loop
    About tangent
    \(\frac{3}{2}MR^2\)
    \(\frac{3}{2}\)
    Circular disc
    Through C.G. and perpendicular to plane
    \(\frac{1}{2}MR^2\)
    \(\frac{1}{2}\)
    Circular disc
    About diameter
    \(\frac{MR^2}{4}\)
    \(\frac{1}{4}\)
    Hollow disc
    Through C.G. and perpendicular to plane
    \(\frac{1}{2}M(R^2+r^2)\)
    \(\frac{1}{2}\)
    Hollow disc
    About diameter
    \(\frac{M(R^2+r^2)}{4}\)
    \(\frac{1}{4}\)
    Solid cylinder
    About geometrical axis
    \(\frac{MR^2}{2}\)
    \(\frac{1}{2}\)
    Solid cylinder
    Through C.G. and perpendicular to geometrical axis
    \(M\left(\frac{L^2}{12}+\frac{R^2}{4}\right)\)
    —
    Hollow cylinder
    About geometrical axis
    \(MR^2\)
    1
    Hollow cylinder
    Through C.G. and perpendicular to geometrical axis
    \(M\left(\frac{L^2}{12}+\frac{R^2}{2}\right)\)
    —
    Hollow sphere
    About diameter
    \(\frac{2}{3}MR^2\)
    \(\frac{2}{3}\)
    Hollow sphere
    About tangent
    \(\frac{5}{3}MR^2\)
    \(\frac{5}{3}\)
    Solid sphere
    About diameter
    \(\frac{2}{5}MR^2\)
    \(\frac{2}{5}\)
    Solid sphere
    About tangent
    \(\frac{7}{5}MR^2\)
    \(\frac{7}{5}\)
    ❖ Special M.I. Points:
    • •
      If body lies on axis of rotation, M.I. about that axis is zero
    • •
      M.I. of cube about diagonal is minimum
    • •
      Internal and external radii of circular lamina \(r\), \(R\): \(I=\frac{M}{2}(R^2+r^2)\)
    • •
      When disc is melted and recast as solid sphere, M.I. about vertical central axis decreases
    • •
      Raw egg has greater M.I. than hard-boiled egg of identical size
    ▢ Kinetic Energy in Rotational Motion:

    Table 1: Kinetic Energy Types

    Motion
    Energy
    Pure translation
    \(K.E._t=\frac{1}{2}mv^2\)
    Pure rotation
    \(K.E._r=\frac{1}{2}I\omega^2=n\cdot\frac{1}{2}mv^2\)
    Rolling without slipping
    \(K.E._T=K.E._t+K.E._r\)
    Total K.E. in rolling
    \(K.E._T=(1+n)\frac{1}{2}mv^2\)
    ❖ Energy Ratio: \(E_t:E_r:E_T=1:n:(1+n)=R^2:K^2:(R^2+K^2)\)
    ❖ Special:
      ◈ _*type: bullet
    1. Rotational energy fraction maximum for ring
    2. Rotational energy fraction minimum for solid sphere
    3. For same mass, radius and angular velocity, rotational K.E. is same if M.I. is same
    ▢ Centre of Mass:
    ❖ Definition: Point where entire mass of body/system can be assumed concentrated
    ❖ Formulae:
    ◉ **table:
      ❖ Properties:
        ◈ **type: bullet
      1. C.M. may or may not lie within material of body
      2. C.M. position is independent of reference frame
      3. C.M. depends on masses and relative positions
      4. C.M. and geometrical centre may not coincide
      5. If \(\vec F*{ext}=0\), then \(\vec p=M\vec v*{cm}=constant\)
      6. Internal forces can change momentum of individual particles but not C.M. momentum
      7. Nature of C.M. motion depends on external force, not internal forces
      8. Sum of moments of masses about C.M. is zero
      9. C.M. describes translatory motion
      ❖ Explosion Cases:
      • •
        Body falling vertically and explodes → C.M. remains on same vertical line
      • •
        Shell moving parabolically and explodes → C.M. continues on same parabolic path
      • •
        If initially particles are at rest → velocity of C.M. remains zero
      ❖ Special Points:
      • •
        Two masses \(m_1>m_2\) connected by light rod → C.M. lies nearer \(m_1\)
      • •
        C.M. of thin triangular plate lies at centroid
      • •
        For stable equilibrium, C.G. should be at lowest position
      • •
        When C.G. rises, body becomes unstable
      ▢ Equilibrium of Rigid Body:

      Table 1: Equilibrium Conditions

      Equilibrium
      Condition
      Translational equilibrium
      \(\sum \vec F=0\)
      Rotational equilibrium
      \(\sum \vec\tau=0\)
      Mechanical equilibrium
      \(\sum \vec F=0\) and \(\sum \vec\tau=0\)
      ▢ Rolling Motion on Inclined Plane:
      ❖ Condition: Body rolls down an inclined plane of inclination \(\theta\) without slipping from rest
      ❖ Let: \(n=\frac{K^2}{R^2}\)

      Table 1: Rolling Down Inclined Plane

      Quantity
      Formula
      Velocity at bottom
      \(v=\sqrt{\frac{2gh}{1+K^2/R^2}}=\frac{v_0}{\sqrt{1+n}}\)
      Acceleration down plane
      \(a=\frac{g\sin\theta}{1+K^2/R^2}=\frac{g\sin\theta}{1+n}=\frac{a_0}{1+n}\)
      Time of descent
      \(t=\sqrt{\frac{2l(1+n)}{g\sin\theta}}=t_0\sqrt{1+n}\)
      Frictional force
      \(f=\frac{K^2}{K^2+R^2}mg\sin\theta=\frac{n}{1+n}mg\sin\theta\)
      Coefficient of rolling friction
      \(\mu=\frac{f}{mg\cos\theta}=\frac{K^2}{K^2+R^2}\tan\theta=\frac{n}{1+n}\tan\theta\)
      Condition for rolling without sliding
      \(\mu>\frac{K^2}{K^2+R^2}\tan\theta=\frac{n}{1+n}\tan\theta\)
      ❖ Notes:
      • •
        \(v_0,a_0,t_0\) are values for smooth sliding body from same height
      • •
        Lower \(K^2/R^2\) → faster body reaches bottom
      • •
        Body cannot roll down smooth inclined plane without friction
      • •
        In pure rolling, no work is done against friction
      • •
        Potential energy converts into translational + rotational kinetic energy
      ❖ Order of Reaching Bottom:

      Table 1: Rolling Bodies from Same Height

      Body
      \(n=K^2/R^2\)
      Speed
      Solid sphere
      \(\frac{2}{5}\)
      Fastest
      Solid cylinder / disc
      \(\frac{1}{2}\)
      Intermediate
      Hollow sphere
      \(\frac{2}{3}\)
      Slower
      Ring / hollow cylinder
      1
      Slowest
      ▢ Falling Rod Pivoted at One End:
      ❖ Condition: Uniform rod of mass \(M\), length \(L\), lower end pivoted, released from vertical position

      Table 1: Rod Falling Formulae

      Quantity
      Formula
      Angular velocity after angle \(\theta\)
      \(\omega=\sqrt{\frac{3g}{L}(1-\cos\theta)}=\sqrt{\frac{6g}{L}}\sin\frac{\theta}{2}\)
      Velocity of free end
      \(v=\omega L=\sqrt{3gL(1-\cos\theta)}=\sqrt{6gL}\sin\frac{\theta}{2}\)
      ▢ Angular Momentum:
      ❖ Formula: \(\vec L=\vec r\times\vec p=I\vec\omega\)
      ❖ Direction: Along axis of rotation
      ❖ Conservation Principle:
      • •
        Ice skater uses conservation of angular momentum
      • •
        Dancer spins faster on folding arms
      • •
        Folding arms decreases M.I., increases angular velocity
      • •
        Angular momentum remains conserved if external torque is zero
      • •
        Sand poured on rotating disc → M.I. increases, angular velocity decreases
      ❖ Examples:
      • •
        Swimmer pulls arms and legs in during dive → M.I. decreases, angular velocity increases
      • •
        If polar ice melts, M.I. of Earth increases, angular velocity decreases, length of day increases
      ▢ Read and Digest:

      Table 1: Important Rotational Motion Points

      Fact
      Point
      Moment of inertia depends on
      Mass and distribution of mass about axis
      C.M. may lie
      Inside or outside material body
      C.M. in rotatory motion
      Location unchanged
      C.M. in translatory motion
      Location changes
      External force absent
      Velocity of C.M. remains constant
      External resultant zero
      Velocity of C.M. constant or zero
      Cycle wheel spokes
      Increase M.I. → smoother and steadier motion
      Flywheel
      Keeps engine speed uniform
      Couple
      Produces purely rotational motion
      Raw egg vs boiled egg M.I.
      Raw egg M.I. greater
      Raw egg and boiled egg spun
      Hard-boiled egg comes to rest earlier
      Ice skater principle
      Conservation of angular momentum
      Gymnast lowers arms on rotating stool
      M.I. decreases
      Dancer folds arms
      M.I. decreases, \(\omega\) increases, K.E. increases, angular momentum conserved
      Sand poured on rotating disc
      \(\omega\) decreases because M.I. increases
      Angular momentum direction
      Along axis of rotation
      Body on axis of rotation
      M.I. about that axis = 0
      Radius of gyration
      Independent of mass
      Polar ice melts
      Length of day increases
      Disc recast into solid sphere
      M.I. decreases
      Fastest rolling body among solid cylinder, solid sphere, hollow sphere
      Solid sphere
      Rotational energy
      Maximum fraction for ring, minimum for solid sphere
      Translational energy
      Maximum fraction for solid sphere, minimum for ring
      Smooth inclined plane
      Body cannot roll without friction
      Pure rolling friction work
      Zero
      Cylinder unrolling by massless thread
      Acceleration = \(\frac{2g}{3}\)
      Rotating water surface
      Parabolic due to centrifugal effect
      Angular speed about opposite end of diameter
      \(\frac{\omega}{2}\)
      Cube M.I.
      Minimum about diagonal
      ▢ Special Rolling Surface Velocity:
      ❖ Condition: Solid sphere rolling without slipping with velocity \(v\)
      ❖ _*table:
        ▢ High-Yield Recall:
        ❖ **table:
          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 enThe 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,
          📅IOM 2013]
          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 sethen 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 seThen 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/seThe 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. speeThe 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:
          📅IOM•BPKIHS•MOE•KU
          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 + be the torque of this force about the origin, then [KU 2015]
          📅KU 2015
          Q76.
          If a gymnast on a rotating stool with his arms outstretched suddenly lower his arms
          📅TOM 2015