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 1: Theorems of Moment of Inertia

Theorem
Formula
Parallel axis theorem
\(I=I_{cm}+mr^2\)
Perpendicular axis theorem
\(I_z=I_x+I_y\)
❖ 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:
  • •
    Rotational energy fraction maximum for ring
  • •
    Rotational energy fraction minimum for solid sphere
  • •
    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 1: Centre of Mass Formulae

Quantity
Formula
Two masses balance condition
\(m_1x_1=m_2x_2\)
C.M. coordinates
\((x,y)=\left(\frac{m_1x_1+m_2x_2}{m_1+m_2},\frac{m_1y_1+m_2y_2}{m_1+m_2}\right)\)
Velocity of C.M.
\(\vec v_{cm}=\frac{m_1\vec v_1+m_2\vec v_2}{m_1+m_2}\)
Acceleration of C.M.
\(\vec a_{cm}=\frac{m_1\vec a_1+m_2\vec a_2}{m_1+m_2}\)
Momentum of system
\(\vec p=M\vec v_{cm}\)
❖ Properties:
  • •
    C.M. may or may not lie within material of body
  • •
    C.M. position is independent of reference frame
  • •
    C.M. depends on masses and relative positions
  • •
    C.M. and geometrical centre may not coincide
  • •
    If \(\vec F_{ext}=0\), then \(\vec p=M\vec v_{cm}=constant\)
  • •
    Internal forces can change momentum of individual particles but not C.M. momentum
  • •
    Nature of C.M. motion depends on external force, not internal forces
  • •
    Sum of moments of masses about C.M. is zero
  • •
    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 1: Velocity of Surface Particle

Point
Formula
Point on surface at angle \(\theta\) above level of C.M.
\(v'=2v\cos\left(45^\circ+\frac{\theta}{2}\right)\)
Point below C.M.
\(v'=2v\cos\left(45^\circ-\frac{\theta}{2}\right)\)
▢ High-Yield Recall:

Table 1: Rotational Motion One-Liners

Fact
Answer
Angular displacement relation
\(\vec s=\vec\theta\times\vec r\)
Angular velocity relation
\(\vec v=\vec\omega\times\vec r\)
Angular acceleration relation
\(\vec a=\vec\alpha\times\vec r\)
Moment of inertia
\(I=mr^2\)
Angular momentum
\(\vec L=\vec r\times\vec p=I\vec\omega\)
Torque
\(\vec\tau=\vec r\times\vec F=I\vec\alpha\)
Rotational K.E.
\(\frac{1}{2}I\omega^2\)
Parallel axis theorem
\(I=I_{cm}+mr^2\)
Perpendicular axis theorem
\(I_z=I_x+I_y\)
Rod about centre
\(\frac{ML^2}{12}\)
Rod about end
\(\frac{ML^2}{3}\)
Ring about central axis
\(MR^2\)
Disc about central axis
\(\frac{1}{2}MR^2\)
Solid sphere about diameter
\(\frac{2}{5}MR^2\)
Hollow sphere about diameter
\(\frac{2}{3}MR^2\)
Solid cylinder about axis
\(\frac{1}{2}MR^2\)
Hollow cylinder about axis
\(MR^2\)
Rolling total K.E.
\((1+n)\frac{1}{2}mv^2\)
Rolling energy ratio
\(E_t:E_r:E_T=1:n:1+n\)
C.M. coordinates
\(x_{cm}=\frac{\sum mx}{\sum m}\)
C.M. velocity
\(\vec v_{cm}=\frac{\sum m\vec v}{\sum m}\)
Translational equilibrium
\(\sum \vec F=0\)
Rotational equilibrium
\(\sum \vec\tau=0\)
Mechanical equilibrium
\(\sum \vec F=0,\ \sum\vec\tau=0\)
Rolling acceleration
\(a=\frac{g\sin\theta}{1+n}\)
Rolling velocity at bottom
\(v=\sqrt{\frac{2gh}{1+n}}\)
Rolling friction
\(f=\frac{n}{1+n}mg\sin\theta\)
Condition for rolling
\(\mu>\frac{n}{1+n}\tan\theta\)
Fastest rolling body
Solid sphere
Slowest rolling body
Ring / hollow cylinder
Couple produces
Pure rotational motion
Ice skater principle
Conservation of angular momentum
Flywheel function
Keeps speed uniform
Cylinder unrolling acceleration
\(\frac{2g}{3}\)
Q1.
The torque due to gravitational force on a body about its centre of mass is:
📅CEE Nursing 2024
Q2.
Physical quantity responsible for angular acceleration is:
📅CEE MBBS 2024B
Q3.
The scientific reason behind placing the handle of a door close to the edge of a door is:
📅CEE MBBS 2024a
Q4.
The centre of mass of two bodies of masses M and m (M > m), separated by distance d:
📅CEE MBBS 2022
Q5.
The centre of mass is not affected by:
📅CEE MBBS 2021
Q6.
A fan starts from rest and completes 10 revolutions in the first 3 seconds with uniform angular acceleration. How many revolutions does it complete in the next 3 seconds?
📅BP 2010
Q7.
What is an important function of the spokes of a bicycle wheel?
📅BP 2011
Q8.
A small object rests in a smooth hemispherical bowl of radius \(0.4\,\mathrm m\), which rotates about its vertical axis. The object is \(0.2\,\mathrm m\) above the bottom. Find the angular speed for equilibrium (\(g=10\,\mathrm{m/s^2}\)).
📅BP 2009
Q9.
A uniform solid billiard ball of mass \(m\) and radius \(r\) is struck with an impulse at a perpendicular distance \(h\) from its centre and acquires speed \(v_0\). Find its angular speed.
📅BP 2009
Q10.
A ring, disc, solid sphere and hollow sphere roll without slipping down the same incline from rest. Which arrives first?
📅MOE 2014
Q11.
A body of mass \(M\) has moment of inertia \(I\) about an axis. Its radius of gyration is:
📅MOE 2014
Q12.
What is the gravitational torque on a rigid body about its centre of mass in a uniform gravitational field?
📅MOE 2014
Q13.
Forces of \(2\,\mathrm N\) and \(4\,\mathrm N\) act downward at opposite ends of a \(0.5\,\mathrm m\) light rod. Where should a third force act for equilibrium?
📅MOE 2011
Q14.
Two particles of masses \(1\,\mathrm{kg}\) and \(2\,\mathrm{kg}\) are separated by \(0.5\,\mathrm m\). Find the centre of mass measured from the 1 kg particle.
📅MOE 2013
Q15.
A uniform circular disc of mass \(2\,\mathrm{kg}\) and radius \(1\,\mathrm m\) has what moment of inertia about a diameter?
📅MOE 2011
Q16.
The moment of inertia of a body about a specified axis does NOT depend on its:
📅MOE 2010
Q17.
An electron of mass \(M\) moves in a circular orbit of radius \(r\). Its classical moment of inertia about the orbital axis is:
📅MOE 2009
Q18.
When a rigid body rolls down an incline without slipping, its lost gravitational potential energy becomes:
📅IE 2011
Q19.
If the net external force on a system is zero, the velocity of its centre of mass:
📅IOM 2013
Q20.
For rotation about a fixed axis with constant moment of inertia, the product of moment of inertia and angular acceleration gives:
📅IOM 2013
Q21.
A solid cylinder of mass \(M\) and radius \(R\) has moment of inertia about its symmetry axis:
📅IOM 2012
Q22.
Two particles move in circles of different radii but complete a revolution in equal times. The ratio of their angular speeds is:
📅KU 2014
Q23.
A uniform heavy disc rotates freely about its central vertical axis. Wax sticks near its rim. Its angular velocity:
📅KU 2012
Q24.
A conducting disc of radius \(R\) rotates with angular speed \(\omega\) in a uniform magnetic field \(B\) perpendicular to its plane. Find the magnitude of emf between its centre and rim.
📅KU 2011
Q25.
Two masses of \(1\,\mathrm{kg}\) and \(2\,\mathrm{kg}\) are separated by \(9\,\mathrm m\). Their centre of mass is how far from the 1 kg mass?
📅Bangladesh 09
Q26.
A spinning disc receives a small amount of wax that sticks to its surface. Neglecting external torque, its angular speed:
📅KU 2009
Q27.
A uniform solid disc of mass \(m\) and radius \(r\) rotates about its symmetry axis with rim speed \(v\). Its kinetic energy is:
📅IOM 2004
Q28.
If Earth’s radius becomes half with its mass and angular momentum unchanged, its rotational period becomes:
📅IOM 2003
Q29.
A solid disc of mass \(M\) rotates about its axis with speed \(v\) at the rim. Find its rotational kinetic energy.
📅IOM 98
Q30.
A uniform solid flywheel of mass \(10\,\mathrm{kg}\) and radius \(0.5\,\mathrm m\) has rotational kinetic energy \(20\,\mathrm J\). Find its angular speed.
📅MOE 2066
Q31.
A constant torque changes angular momentum from \(L_0\) to \(4L_0\) in \(3\,\mathrm s\). Find the torque.
📅MOE 2008
Q32.
The rotational kinetic energy of a rigid body of moment of inertia \(I\) spinning at angular speed \(\omega\) is:
📅MOE 2065
Q33.
A wheel starts from rest with angular acceleration \(\alpha=6t\,\mathrm{rad/s^2}\). Find the time required for 10 revolutions.
Q34.
Angular momentum changes from \(2\,\mathrm{kg\,m^2/s}\) to \(4\,\mathrm{kg\,m^2/s}\) in \(4\,\mathrm s\). Find the average torque.
📅TE-04)
Q35.
If net external torque on a system is zero, which quantity is conserved?
📅TE-051
Q36.
A shell initially at rest explodes into fragments. Neglecting external forces, the centre of mass of the fragments:
📅IE-08•BP 2017
Q37.
A uniform disc of mass \(M\) and radius \(R\) rotates about a tangent axis lying in the plane of the disc and parallel to a diameter. Its moment of inertia is:
📅BPKIHS-08
Q38.
A rotating body has mass \(m\), radius of gyration \(k\), and angular acceleration \(\alpha\). The torque is:
Q39.
What is the ratio of the angular speeds of the minute and hour hands of a clock?
📅BPKIHS 2007
Q40.
The centre of gravity of a rigid body:
📅BPKIHS 1994
Q41.
A uniform rod of length \(L\) has radius of gyration about an axis through its midpoint and perpendicular to its length equal to:
📅BPKIHS 1994
Q42.
Two homogeneous bodies of identical shape and dimensions are made from aluminium and iron. Their moments of inertia about corresponding axes satisfy:
📅BPKIHS-95
Q43.
Three point masses \(m\) occupy the vertices of an equilateral triangle of side \(L\). Find the moment of inertia about an axis along one side.
📅BPKIHS-96
Q44.
An explosive shell follows a parabolic trajectory and explodes in flight. Ignoring air resistance, its fragments’ centre of mass:
📅BPKIHS-97)
Q45.
Find the ratio of the angular speed of Earth’s axial rotation to that of the hour hand of a clock.
📅BPKIHS 1999
Q46.
If the radius of Earth’s orbit around the Sun were reduced to one-fourth, the orbital period would become:
📅BPKIHS 2000
Q47.
Moment of inertia of a thin circular ring of mass \(M\) and radius \(R\) about a diameter is:
Q48.
The moment of inertia of a uniform thin rod about an axis through one end, perpendicular to its length \(L\), is:
Q49.
Moment of inertia of a solid uniform sphere about its diameter is:
Q50.
A solid cylinder of radius \(R\) and length \(L\) has equal moments of inertia about its symmetry axis and an equatorial diameter. Find \(R/L\).
Q51.
A uniform disc of moment of inertia \(I_0\) about its symmetry axis is recast into a thin ring of the same mass and radius. Its moment of inertia about a diameter is:
Q52.
A uniform disc has moment of inertia \(I_0\) about a diameter. Its moment of inertia about an axis through its rim, perpendicular to its plane, is:
Q53.
The radii of two homogeneous steel spheres are \(R\) and \(2R\). Their moments of inertia about diameters are in the ratio:
📅KU 2009
Q54.
A circular disc of radius \(R\) and mass \(M\) has a small disc of radius \(R/2\) removed so that the hole touches the original rim. Find the remaining moment of inertia about the original centre, perpendicular to the plane.
Q55.
A thin uniform rod of mass \(M\), length \(L\) spins about its centre with two beads of mass \(m\) initially at its centre. The beads move to opposite ends. Find the final angular speed in terms of initial \(\omega_0\).
Q56.
Three identical thin rods, each mass \(M\) and length \(L\), lie along the positive coordinate axes, each starting at the origin. Find their total moment of inertia about the \(z\)-axis.
Q57.
A solid sphere and a thin hollow sphere of equal mass have the same moment of inertia about their diameters. Their radii, solid to hollow, are in the ratio:
Q58.
Two uniform discs have the same mass and thickness but different densities \(\rho_1\) and \(\rho_2\). Their moments of inertia about symmetry axes satisfy \(I_1:I_2=\):
Q59.
A wheel of moment of inertia \(5\times10^{-3}\,\mathrm{kg\,m^2}\) rotates at 20 revolutions per second. Find the magnitude of constant torque required to stop it in 10 s.
Q60.
A thin hollow cylinder moves at speed \(v\), once sliding without rotation and once rolling without slipping. Find \(K_{sliding}:K_{rolling}\).
Q61.
A solid sphere of mass \(M\) rolls without slipping with centre-of-mass speed \(v\). Its total kinetic energy is:
Q62.
A solid sphere of mass \(M\) rotates about its diameter. An equatorial point has speed \(v\). Its rotational kinetic energy is:
Q63.
A rigid body rolls without slipping with rotational kinetic energy equal to 40% of its total kinetic energy. It is a:
Q64.
For a solid sphere rolling without slipping, the ratio of rotational kinetic energy to total kinetic energy is:
Q65.
Find the minimum coefficient of static friction for a solid cylinder to roll without slipping down an incline at angle \(\theta\).
Q66.
A wheel has mass \(10\,\mathrm{kg}\) and moment of inertia \(160\,\mathrm{kg\,m^2}\). Find its radius of gyration.
Q67.
A solid disc of radius \(50\,\mathrm{cm}\) rotates about its central symmetry axis. Find its radius of gyration.
Q68.
A uniform disc of mass \(20\,\mathrm g\) rotates at \(30\,\mathrm{rpm}\). When \(20\,\mathrm g\) of sand lands at a distance \(5\,\mathrm{cm}\) from the axis, its speed becomes \(24\,\mathrm{rpm}\). Find its radius.
Q69.
A particle has angular momentum \(L\). If its frequency doubles and kinetic energy becomes half, its new angular momentum is:
Q70.
Constant torque changes angular momentum from \(L_0\) to \(4L_0\) in 4 seconds. Find its magnitude.
📅IOM•BPKIHS•MOE•KU
Q71.
A flywheel initially at \(300\,\mathrm{rpm}\) decelerates uniformly to rest in 10 s. How many revolutions does it complete?
Q72.
A solid sphere of mass \(2\,\mathrm{kg}\), radius \(5\,\mathrm{cm}\), initially rotating at \(300\,\mathrm{rpm}\), is brought to rest in \(2\pi\) revolutions. Find the magnitude of retarding torque.
Q73.
Two particles initially at rest move toward each other under mutual attraction, with no external force. Their centre-of-mass velocity is:
Q74.
Two point masses \(m_1\) and \(m_2\) are separated by distance \(x\). Find the centre-of-mass distance from \(m_2\).
Q75.
Two particles attract each other by internal forces in the absence of net external force. What is the displacement of their common centre of mass if it is initially at rest?
Q76.
Two blocks of masses \(5\,\mathrm{kg}\) and \(2\,\mathrm{kg}\) are joined by a spring on a frictionless surface. The 5 kg block is given a velocity of \(14\,\mathrm{m/s}\); the other is initially at rest. Find centre-of-mass speed.
Q77.
A projectile has initial speed \(u\) at angle \(\theta\). At its highest point it explodes into two equal fragments. One comes to rest immediately. Where does the other land relative to the launch point (same ground level)?
Q78.
A uniform disc of diameter \(56\,\mathrm{cm}\) has a circular hole of diameter \(42\,\mathrm{cm}\) cut from its edge, tangential to the rim. Find the shift of its centre of mass.
Q79.
Two masses of \(1\,\mathrm{kg}\) and \(2\,\mathrm{kg}\) are 9 m apart. Find their centre of mass measured from the 1 kg mass.
📅MOE
Q80.
A force \(\vec F\) acts at position vector \(\vec r\), producing torque \(\vec\tau=\vec r\times\vec F\). Which statement is true?
📅KU 2015
Q81.
A gymnast sitting on a rotating stool pulls their outstretched arms closer to the rotation axis. Which quantity decreases?
📅IOM 2015