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NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET

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MULTIPLE CHOICE QUESTIONS I

Q.1. For which of the following does the centre of mass lie outside the body ?
(a) A pencil
(b) A shotput
(c) A dice
(d) A bangle
Ans. (d)
Solution.
A bangle is a ring like shape and the centre of mass of ring lies at its centre which is outside the ring or bangle. Hence verifies the option (d).

Q.2. Which of the following points is the likely position of the centre of mass of the system shown in Figure ?
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET(a) A
(b) B
(c) C
(d) D
Ans. 
(c)
Solution.

The position of centre of mass of the system in this problem is closer to heavier mass or masses or we can say that it depends upon distribution of mass. So it is likely to be at C. In the above diagram, lower part of the sphere containing sand is more heavier than upper part containing air. Hence CM of the system lies below the horizontal diameter.
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
Q.3. A particle of mass m is moving in yz-plane with a uniform velocity v with its trajectory running parallel to +ve y-axis and intersecting z-axis at z = a (Figure). The change in its angular momentum about the origin as it bounces elastically from a wall at y = constant is:
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET(a) mva êx
(b) 2mva êx
(c) ymv êx
(d) 2ymv êx
Ans. 
(b)
Solution.
 Angular momentum is an axial vector, i.e. always directed perpendicular to the plane of rotation and along the axis of rotation.
In cartesian co-ordinates if NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
ThenNCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
KE of the system remains conserved, in elastic collision. So, the ball will bounce back with the same speed v but in opposite direction, i.e. along -ve y-axis.
The initial velocity isNCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEETand after reflection from the wall, the final velocity is NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET. The trajectory is described as position vector NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET.
Hence, the change in angular momentum is
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET

Q.4. When a disc rotates with uniform angular velocity, which of the following is not true?

(a) The sense of rotation remains same.
(b) The orientation of the axis of rotation remains same.
(c ) The speed of rotation is non-zero and remains same.
(d) The angular acceleration is non-zero and remains same.
Ans. 
(d)
Solution.

The rate of change of angular velocity is defined as angular acceleration. If particle has angular velocity ω1 at time
t1, and angular velocity ωat time t, then
Angular acceleration α = ω– ω1 / t– t1
When the disc is rotated with constant angular velocity, angular acceleration of the disc is zero. Because we know that angular acceleration
α  = ∆ ω/∆ t
Here ω is constant, so ∆ ω = 0

Q.5. A uniform square plate has a small piece Q of an irregular shape removed and glued to the centre of the plate leaving a hole behind (Figure). The moment of inertia about the z-axis is then

NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET(a) Increased
(b) Decreased
(c) The same
(d) Changed in unpredicted manner
Ans. 
(b)
Solution.

After removing the matter from Q it is stick at P through axis of rotation passes, but axis of rotation does not passes through Q. So comes closer to the axis of rotation.

Q.6. In previous problem , the CM of the plate is now in the following quadrant of x-y plane,
(a) I
(b) II
(c) III
(d) IV
Ans. 
(c)
Solution.

Let us consider the diagram below, which shows the position of the piece which is removed from the plate. First center of mass is at the centre of the plate (only if its mass is uniformly distributed over the surface) when the piece is removed from quadrant I, therefore the centre of mass is shifted to the  opposite of the quadrant III.

NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET

Position of CM is shown by point  X in the diagram.

Q.7. The density of a non-uniform rod of length 1m is given by
ρ(x) = a(1+bx2)
where a and b are constants and o≤x≤1.
The centre of mass of the rod will be at
(a)NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
(b)NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
(c)NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
(d)NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
Ans. 
(a)
Solution.

ρ(x) = a(1+bx2)
At b 0, p(x) = a i.e./ p is constant in this case C.M. must lies at mid-point of 1 m
i.e., at x = 0.5 m ,by substituting value of b = 0 in option (a), (c), d).
We observe that the C.M. lies at 0.5 m at b = 0.
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEETHence, option (a) verifies.

Q.8. A Merry-go-round, made of a ring-like platform of radius R and mass M, is revolving with angular speed ω . A person of mass M is standing on it. At one instant, the person jumps off the round, radially away from the centre of the round (as seen from the round).
The speed of the round afterwards is
(a) 2ω
(b) ω
(c) ω/2
(d) 0
Ans.
(b)
Solution.
As no torque is exerted by the person jumping, radially away from the centre of the round (as seen from the round), let the total moment of inertia of the system is 2I (round + Person (because the total mass is 2M) and the round is revolving with angular speed ω Since the angular momentum of the person when it jumps off the round is Iω the actual momentum of round seen from ground is  2Iω – Iω = Iω
So we conclude that the angular speed remains same, i.e ω
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET


MULTIPLE CHOICE QUESTIONS II

Q.1. Choose the correct alternatives:
(a) For a general rotational motion, angular momentum L and angular velocity ω need not be parallel.
(b) For a rotational motion about a fixed axis, angular momentum L and angular velocity ω are always parallel.
(c) For a general translational motion, momentum p and velocity v are always parallel.
(d) For a general translational motion, acceleration a and velocity v are always parallel.
Ans. (a, c)
Solution.

(a) For a general rotational motion where axis of rotation is not symmetric. Angular momentum Z and angular velocity 0) need not be parallel. The wobbly motion of a wheel rotating about an axis inclined at a small angle to the symmetry axis of the wheel represents a situation where angular momentum and angular velocity are not parallel.
(b) Fixed axis should pass through CM of the body, so it is not necessary angular momentum Z and angular velocity ω are always parallel.
(c) As we know in a general translational motion linear momentum is given by, p = mv , hence, direction of p is always along v .
(d) In projectile motion, v and a are not always parallel.

Q.2. Figure shows two identical particles 1 and 2, each of mass m, moving in opposite directions with same speed v along parallel lines.
At a particular instant, r1 and r2 are their respective position vectors drawn from point A which is in the plane of the parallel lines.

NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEETChoose the correct options:
(a) Angular momentum l1 of particle 1 about A is l1 = mvd1
(b) Angular momentum l2 of particle 2 about A is l2 = mvr2
(c) Total angular momentum of the system about A is l = mv(r1 + r2)
(d) Total angular momentum of the system about A isl = mv(d2 − d1)

⊗ represents a  unit vector coming out of the page.
⊗ represents a  unit vector going into the page.
Ans. (a,b)
Solution.

The angular momentum L of a particle about the selected axis o f rotation is defined to be NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET where NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET is the position vector of the particle and NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET is the linear momentum. The direction of  NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET is perpendicular to both  NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET  by right hand thumb rule or screw rule.The angular momentum about point A, For particle 1.

NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET

Similarly, for particle 2,

NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET(∴ r2 sinθ2 = d2) (into the plane of the paper)
∴ Total angular momentumNCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
Hence (a) and (d) are correct options.

Q.3. The net external torque on a system of particles about an axis is zero. Which of the following are compatible with it ?
(a) The forces may be acting radially from a point on the axis.
(b) The forces may be acting on the axis of rotation.
(c) The forces may be acting parallel to the axis of rotation.
(d) The torque caused by some forces may be equal and opposite to that caused by other forces.
Ans. 
(a, b, c, d) 
Solution.
Torque on a system of particles about the axis of rotation is defined as NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET............ (i)
where, θ is angle between NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET , and NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET is a unit vector along the direction of torque and which is perpendicular to both NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET.NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
whereNCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEETis the component of F perpendicular to r.
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEETOR
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
whereNCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEETis the perpendicular distance from the origin to the line of action of the force. It is also called the lever arm.
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET(a) If the net external torque on a system of particles about an axis is zero, then NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET. When forces act radially, θ = 0 henceNCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET, so option (a) is correct.
(b) When forces arc acting on the axis of rotation, r = 0,NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
(c) When forces acting parallel to the axis of rotation θ = 0°,NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
(d) When torque by forces are equal and opposite,NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
Important point: To get the direction where you can use right hand rule: Place the fingers of right hand along r and then curl them into F through the smaller angle between them, n is directed along the (stretched) thumb. For involving two dimensions only, can be replaced by sense of rotation, clockwise or anticlockwise; if the fingers of the right hand curl (while going from clockwise, torque is taken as clockwise and when they curl anticlockwise torque is taken as anticlockwise.)

Q.4. Figure shows a lamina in x-y plane. Two axes z and z′ pass perpendicular to its plane. A force F acts in the plane of lamina at point P as shown. Which of the following are true? (The point P is closer to z′-axis than the z-axis.)
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET(a) Torque τ caused by F about z axis is along -kˆ.
(b) Torque τ′ caused by F about z′ axis is along -kˆ .
(c) Torque τ caused by F about z axis is greater in magnitude than that about z axis.
(d) Total torque is given be τ = τ + τ′.
Ans. 
(b,c)
Solution.

As we know that
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
And direction of z is perpendicular to plane of r and F by right hand thumb rule.
(a) Torque caused by force F about z-axis is in +k direction by right hand thumb rule.
(b) Similarly Torque caused by force F about z´-axis is in –k direction.
(c) r>r´ and θ>θ´
So, rFsinθ > r´Fsinθ´
Hence (d) There is no sense in adding torques about 2 different axes.

Q.5. With reference to Figure of a cube of edge a and mass m, State whether the following are true or false. (O is the centre of the cube.)
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET(a)  The moment of inertia of cube about z-axis is Iz = Ix + Iy
(b) The moment of inertia of cube about z′ is I′z = Iz +NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
(c) The moment of inertia of cube about z″ is = Iz + NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
(d) Ix = Iy
Ans. 
(b,d)
Solution.

Option a is false since perpendicular axes theorem is applicable only for laminar/planar objects.
Distance between z and z′ =NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
Applying parallel axis theorem
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
So option b is correct.
Z′′ and z are skew lines. That is, they are neither parallel nor perpendicular. So option c is incorrect as it is using parallel axis theorem.
Since the cube is symmetric, moment of inertia about x and y axis are equal. So, correct options are b and d.


VERY SHORT ANSWER TYPE QUESTIONS

Q.1. The centre of gravity of a body on the earth coincides with its centre of mass for a ‘small’ object whereas for an ‘extended’ object it may not. What is the qualitative meaning of ‘small’ and ‘extended’ in this regard?
For which of the following the two coincides? A building, a pond, a lake, a mountain?
Ans. 
Centre of gravity is the point through which the resultant of system of forces (due to weight) of all the particles constituting the body passes for all positions of body.
Centre of mass is a point where whole mass of a body can be concentrated.
When the vertical height of the object is very small as compared to earth’s radius, we call the object small otherwise it is extended.
(a) Building and pond are small object so their centre of mass and centre of gravity coincide.
(b) Deep lake and mountain are extended objects so their centre of mass doe of mass does not coincide with centre of gravity.

Q.2. Why does a solid sphere have smaller moment of inertia than a hollow cylinder of same mass and radius, about an axis passing through their axes of symmetry?
Ans.
As we know that
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
And all the mass in cylinder lies a distance R from axis of symmetry but most of the mass of solid sphere lies at smaller distance than R.
Therefore
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET

Q.3. The variation of angular position θ, of a point on a rotating rigid body, with time t is shown in Figure. Is the body rotating clock-wise or anti-clockwise?
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEETAns. 
According to question slope of θ-t graph is positive, i.e
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
And therefore slope indicate anti-clockwise which is traditionally taken as positive.

Q.4. A uniform cube of mass m and side a is placed on a frictionless horizontal surface. A vertical force F is applied to the edge as shown in Figure. Match the following (most appropriate choice):
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET

 mg/4 < F < mg /2 Cube will move up
 F > mg/2  Cube will not exhibit motion
 F > mg  Cube will begin to rotate and slip at A
  F = mg/4     Normal reaction effectively at a/3 from A, no motion        

Ans. From the problem
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET= Torque due to F at A = aF
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET=  Torque due to mg at A = amg/2
Cube will not move if NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET=NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
So a-(ii)
Cube will rotate if NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
So b-(iii) 
If F>mg
Then cube will move up.
So, c-(i)
When the normal is a/3 from A.
So, for no rotation we balance the torque
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
Balancing force
N = mg - F
Solving the above two equations
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
So, d-(iv) 


Q.5. A uniform sphere of mass m and radius R is placed on a rough horizontal surface (Figure). The sphere is struck horizontally at a height h from the floor.
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEETMatch the following:

 h = R/2 Sphere rolls without slipping with a constant velocity and no loss of energy.
 h = R Sphere spins clockwise, loses energy by friction.
 h = 3R/2 Sphere spins anti-clockwise, loses energy by friction.
 h = 7R/5 Sphere has only a translational motion, looses  energy by friction.

Ans.
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEETApplying conservation of angular momentum just before and after collision.
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
And as there is rolling without slipping
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
⇒ h - R = 2R/5
⇒ h = 7R/5
So, sphere will roll with slipping (no loss of energy) when h =7R/5
So, d-(i)
Torque due to applied force about centre of mass
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
If h=R thenNCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
Therefore, sphere will only have translational motion and loose energy by friction.
So, b-(iv)
If h>R, then torque is positive. Hence sphere will spin clockwise.
So, c-(ii)
If h<R, then torque is negative. Hence sphere will spin anticlockwise.
So, a-(iii)

Q.6. The vector sum of a system of non-collinear forces acting on a rigid body is given to be non-zero. If the vector sum of all the torques due to the system of forces about a certain point is found to be zero, does this mean that it is necessarily zero about any arbitrary point?
Ans. No, The sum of all the torques about the given certain point o is zero.
So,NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
Now, for any other arbitrary point sum of torques can be written as
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
In the above expression the second term may or may not vanish.

Q.7. A wheel in uniform motion about an axis passing through its centre and perpendicular to its plane is considered to be in mechanical (translational plus rotational) equilibrium because no net external force or torque is required to sustain its motion. However, the particles that constitute the wheel do experience a centripetal acceleration directed towards the centre. How do you reconcile this fact with the wheel being in equilibrium?
How would you set a half-wheel into uniform motion about an axis passing through the centre of mass of the wheel and perpendicular to its plane? Will you require external forces to sustain the motion?
Ans. 
As the system is the symmetrical system, the centripetal acceleration in a wheel arise due to internal elastic which exists in pair and cancel each other.
In the second part of question, in half wheel the distribution of mass about centre of mass is not symmetrical.
Therefore, the direction of angular momentum doesn’t coincide with the direction of angular velocity and hence an external torque is required to maintain rotation.

Q.8. A door is hinged at one end and is free to rotate about a vertical axis (Figure). Does its weight cause any torque about this axis? Give reason for your answer.
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEETAns. 
No, As we know thatNCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
So force can be produce torque only along the direction normal to itself. So, when the door in x-y plane, the torque produced by gravity can only be along be positive or negative z direction, never about an axis passing through y direction.

Q.9. (n-1) equal point masses each of mass m are placed at the vertices of a regular n-polygon. The vacant vertex has a position vector a with respect to the centre of  the polygon. Find the position vector of centre of mass.
Ans.
As we know that position of centre of mass is
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
Since the centre of mass of regular n-polygon lies in the geometrical centre.
And n-1 equal point masses each of mass m are placed at n-1 vertices of n polygon. Therefore, it’s centre of mass will be
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
where b = distance of n-1 masses from centre of mass
(n-1)mb + ma = 0

b = -a/(n-1)
Negative sign indicates that isNCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEETin direction opposite to NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET


LONG ANSWER TYPE QUESTIONS

Q.1. Find the centre of mass of a uniform
(a) half-disc,
(b) quarter-disc.
Ans. 
(a) Let M be the mass of half disc and R be its radius.
Mass per unit lengthNCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
Let cut a semi-circular ring out of this semi disc at distance r and r+dr from centre.
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEETSo, Area of elementNCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
Mass of elementary ring = dm =NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
For this semi-circular ring centre of mass isNCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET

So,NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
Hence centre of mass of semi-circular disc =NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
(b) Let M be the mass of quarter disc and R be its radius.
Mass per unit lengthNCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
Similarly, as above part
Area of element =NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
Mass of elementary ring = dm =NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
For this semi-circular ring centre of mass isNCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
So,
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
Hence centre of mass of Quarter disc =NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET

Q.2. Two discs of moments of inertia I1 and I2 about their respective axes (normal to the disc and passing through the centre), and rotating with angular speed ω1 andω2 are brought into contact face to face with their axes of rotation coincident.
(a) Does the law of conservation of angular momentum apply to the situation? why?
(b) Find the angular speed of the two-disc system.
(c) Calculate the loss in kinetic energy of the system in the process.
(d) Account for this loss.

Ans. 
(a) Yes, since there is no external torque on system.
All external forces act through axis of rotation, hence produce no torque.
(b) When two of them coincide let ω be their common angular velocity.
Applying conservation of angular momentum

Final angular momentum= Initial angular momentum
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
(c) NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
Change in kinetic energy
=NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET 
(d) The loss of kinetic energy is due to work against friction between two discs.

Q.3. A disc of radius R is rotating with an angular speed ωo about a horizontal axis. It is placed on a horizontal table. The coefficient of kinetic friction is µk.
(a) What was the velocity of its centre of mass before being brought in contact with the table?
(b) What happens to the linear velocity of a point on its rim when placed in contact with the table?
(c) What happens to the linear speed of the centre of mass when disc is placed in contact with the table?
(d) Which force is responsible for the effects in (b) and (c).
(e) What condition should be satisfied for rolling to begin?
(f) Calculate the time taken for the rolling to begin.
Ans. 
(a) As from the question the disc is rotating about its horizontal axis before coming in contact.
Hence it’s   Vcom = 0
(b) Linear velocity of point at rim decreases due to force of friction.
(c) Linear speed of centre of mas of disc increases due to acceleration gained by it due to friction.

(d) Friction is responsible for effects in b and C.
(e) Rolling starts when Vcm = ωR
where ω is angular speed of disc.
(f) Acceleration produced in centre of mass due to friction
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
Angular acceleration produced by torque due to friction
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
As we know that
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
For rolling without slipping
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
So, NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET


Q.4. Two cylindrical hollow drums of radii R and 2R, and of a common height h, are rotating with angular velocities ω (anti-clockwise) and ω (clockwise), respectively. Their axes, fixed are parallel and in a horizontal plane separated by (3R + δ ) . They are now brought in contact (δ → 0).
(a) Show the frictional forces just after contact.
(b) Identify forces and torques external to the system just after contact.
(c) What would be the ratio of final angular velocities when friction ceases?
Ans. 
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEETHere F and F′ are external forces through support.
Fnet = 0
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
Let ω1 and ω2 be final angular velocities (anticlockwise and clockwise respectively)
Since there will be no friction both drum has same linear velocity.
Hence, VA = VB
1 = Rω2
So,NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET

Q.5. A uniform square plate S (side c) and a uniform rectangular plate R (sides b, a) have identical areas and masses (Figure).

NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEETNCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEETShow that
 (i) IxR / IxS < 1;
(ii) IyR/ IyS > 1;
(iii) IZR/ IZs > 1.
Ans. According to question
Area of square=Area of rectangular plate

c2 = a x b
(a) As I ∝ area2
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET(as from diagram b<c)
(b)NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET(as from diagram a>c)
(c) NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET

NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
So, NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET

Q.6. A uniform disc of radius R, is resting on a table on its rim.The coefficient of friction between disc and table is µ (Figure). Now the disc is pulled with a force F as shown in the figure. What is the maximum value of F for which the disc rolls without slipping?
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
Ans. According to question, let
f = force of friction
F = applied force
a = acceleration produced
α = angular acceleration
So,
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
NCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
⇒ f = Ma/2
Solving above equations

We get f = F/3
AndNCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
ThereforeNCERT Exemplar: System of Particles & Rotational Motion - Notes | Study Physics Class 11 - NEET
⇒ F ≤ 3μmg

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