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Linear Momentum and Angular Momentum | Physics Class 11 - NEET PDF Download

Angular Displacement

It is the angle described by the position vector Linear Momentum and Angular Momentum | Physics Class 11 - NEETabout the axis of rotation.

Angular DisplacementAngular Displacement

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

i.e., angular displacement is a vector quantity whose direction is given by the right-hand rule. It is also known as an axial vector. For anti-clockwise rotation, the direction of θ\thetaθ is perpendicular to the plane, outward along the axis of rotation, and vice-versa.

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

If a body rotates about a fixed axis then all the particles will have same angular displacement ( although linear displacement will differ from particle to particle in accordance with the distance of particles from the axis of rotation).

Angular Velocity

The angular displacement per unit time is defined as angular velocity.

Angular VelocityAngular Velocity

  • Linear Momentum and Angular Momentum | Physics Class 11 - NEET
  • Linear Momentum and Angular Momentum | Physics Class 11 - NEET

ω is an axial vector, whose direction is normal to the rotational plane and its direction is given by the right-hand screw rule.

  • Linear Momentum and Angular Momentum | Physics Class 11 - NEET
  • The magnitude of an angular velocity is called the angular speed, which is also represented by ω.

Angular Acceleration

The rate of change of angular velocity is defined as angular acceleration.

If a particle has angular velocity ωat time t1 and angular velocity ω2 at time t2, then:

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

It is an axial vector whose direction is along the change in direction of angular velocity, normal to the rotational plane, outward or inward along the axis of rotation (depending upon the sense of rotation).

Equations of Linear Motion and Rotational Motion

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Question for Linear Momentum and Angular Momentum
Try yourself:
Which quantity is defined as the angle described by the position vector about the axis of rotation?
View Solution

Linear Momentum 

Linear momentum is a product of the mass (m) of an object and the velocity (v) of the object. If an object has higher momentum, then it harder to stop it. 

The formula for linear momentum is p = mv. 

The overall amount of momentum stays the same. This idea is known as the conservation of momentum.

Linear Momentum of System of Particles

We know that the linear momentum of the particle is
p = mv
Newton’s second law for a single particle is given by,
Linear Momentum and Angular Momentum | Physics Class 11 - NEET
where F is the force of the particle. For ‘ n ‘ no. of particles total linear  momentum is,
P = p+ p+…..+pn
each of momentum is written as  mv+ m2v+ ………..+ mnvn. We know that velocity of the centre of mass is
Linear Momentum and Angular Momentum | Physics Class 11 - NEET
mv =  Σ  mivi
So comparing these equations we get,
P = M V
Therefore we can say that the total linear momentum of a system of particles is equal to the product of the total mass of the system and the velocity of its center of mass. Differentiating the above equation we get,
Linear Momentum and Angular Momentum | Physics Class 11 - NEET
dv/dt is acceleration of centre of mass, MA is the force external. So,
Linear Momentum and Angular Momentum | Physics Class 11 - NEET
This above equation is nothing but Newton’s second law to a system of particles. If the total external force acting on the system is zero,
Linear Momentum and Angular Momentum | Physics Class 11 - NEET
This means that P = constant. So whenever the total force acting on the system of a particle is equal to zero then the total linear momentum of the system is constant or conserved. This is nothing but the law of conservation of total linear momentum of a system of particles. 

Conservation of Total Linear Momentum of a System of Particles

  • Radioactive Decay  is a process where an unstable nucleus splits up in relatively stable nuclei releasing a huge amount of energy. 
  •  Suppose there is a parent nucleus which is unstable and it wants to become stable, in order to attain stability it will emit α particle and another daughter nucleus. 
  •  This daughter nucleus is much more stable than the parent nucleus. This what radioactive decay is. Now suppose the parent nucleus is at rest and also the mass of the α is m and the daughter nucleus is M
  •  So the mass of the parent nucleus will be m + M. Here everything that is happening is not due to the external force but all that happens is due to the internal force. So here Fext = 0, we can say that. 
  •  We know the concept of torque as a turning effect of force. Torque is regarded as a rotational analog of force. 
  •  According to Newton's second law of motion, force is equal to the rate of change of linear momentum. 
  •  Therefore, if we can represent torque as the rate of change of some quantity, that quantity would be the rotational analog of linear momentum, and it can be called angular momentum, which we will study in detail in this document. 

Angular Momentum 

If p is the linear momentum of a particle in a given reference frame, then the angular momentum of the particle about an origin O in this reference frame is defined as

Linear Momentum and Angular Momentum | Physics Class 11 - NEET Linear Momentum and Angular Momentum | Physics Class 11 - NEET = position vector of the particle with respect to the point about which angular momentum is to be calculated.

θ = angle between vectors r and p

Linear Momentum and Angular Momentum | Physics Class 11 - NEET = perpendicular distance of the line of motion of particle from point O.

Linear Momentum and Angular Momentum | Physics Class 11 - NEET = perpendicular component of momentum.

SI unit of angular momentum is kg ms-1.

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Examples of Angular Momentum

  • The angular velocity of a planet around the Sun increases when it comes near the Sun. When a planet revolving around the Sun in an elliptical orbit comes near the Sun, the moment of inertia of the planet about the Sun decreases. In order to conserve angular momentum, the angular velocity shall increase. Similarly, when the planet is away from the Sun, there will be a decrease in the angular velocity.
  • The ice skater and the ballet dancer increase or decrease the angular velocity of spin about a vertical axis by pulling or extending out their limbs.

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

  • A : axis is horizontal; angular momentum about vertical axis = 0.
    B : axis is vertical; angular momentum about vertical axis is still zero; man and chair spin in direction opposite to spin of the wheel.

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Torque or Moment of Force

When a body that is hinged, suspended, or pivoted starts to rotate due to the action of a force, the force is said to exert a torque on the body.

Torque, or the moment of force around a rotational axis, is determined by multiplying the force with the perpendicular distance from the axis of rotation to the line of action of the force.

The formula for calculating the magnitude of torque is: Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Units:

  • In the M.K.S. system: Newton-meter (N·m)
  • In the C.G.S. system: dyne-centimeter (dyne·cm)

Dimension: Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Vector form: Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Relation between Torque and Angular Momentum

As we know that,

Differentiating with respect to the time we get,

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Couple

A couple refers to a specific combination of forces that can cause a body to rotate even when it is free to move. This is known as a couple of forces.

  • A couple is defined as a pair of equal but opposite forces that do not act along the same line. The effect of a couple is measured by its moment of couple, or torque, which can be expressed as:Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

  • Both torque and couple generally represent the same concept. However, the key difference lies in how the forces act: in a couple, both forces are applied externally, whereas, in torque, one force is external while the other is a reactionary force.
  • The work done by torque when twisting a wire can be given as:

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Q1. A particle of mass m is moving along the line y = b, z = 0 with constant speed v. State whether the angular momentum of the particle about the origin is increasing, decreasing, or constant. 

Sol:

Linear Momentum and Angular Momentum | Physics Class 11 - NEETLinear Momentum and Angular Momentum | Physics Class 11 - NEET constant as m, v, and b all are constants.

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Direction of Linear Momentum and Angular Momentum | Physics Class 11 - NEET also remains the same. Therefore, the angular momentum of the particle about the origin remains constant with due course of time.

Note: In this problem Linear Momentum and Angular Momentum | Physics Class 11 - NEET is increasing, q is decreasing but r sin q, i.e., b remains constant. Hence, the angular momentum remains constant.

Q2. A particle of mass m is projected with velocity v at an angle q with the horizontal. Find its angular momentum about the point of projection when it is at the highest point of its trajectory. 

Sol: At the highest point, it has only horizontal velocity vx = v cos q. The length of the perpendicular to the horizontal velocity from 'O' is the maximum height, where

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

⇒ Angular momentum L = Linear Momentum and Angular Momentum | Physics Class 11 - NEET             Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Angular Momentum of a Rigid Body Rotating About a Fixed Axis 

  • Suppose a particle P of mass m is going in a circle of radius r and at some instant the speed of the particle is v. 
  • For finding the angular momentum of the particle about the axis of rotation, the origin may be chosen anywhere on the axis. 
  • We choose it at the center of the circle. In this case Linear Momentum and Angular Momentum | Physics Class 11 - NEET  are perpendicular to each other Linear Momentum and Angular Momentum | Physics Class 11 - NEETand are along the axis. Thus, the component along the axis is mvr itself. 
  • The angular momentum of the whole rigid body about AB is the sum of components of all particles, i.e., L = Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Here, I is the moment of inertia of the rigid body about AB.

Note: Angular momentum about the axis is the component of Linear Momentum and Angular Momentum | Physics Class 11 - NEET along the axis. In most of the cases, angular momentum about axis is Iω.

Q3. Two small balls A and B, each of mass m, are attached rigidly to the ends of a light rod of length d. The structure rotates about the perpendicular bisector of the rod at an angular speed w. Calculate the angular momentum of the individual balls and of the system about the axis of rotation.

Sol:

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Consider the situation shown in the figure. The velocity of the ball A with respect to the center O is 

v = Linear Momentum and Angular Momentum | Physics Class 11 - NEET.

The angular momentum of the ball with respect to the axis is

L1 = mvr = Linear Momentum and Angular Momentum | Physics Class 11 - NEET = Linear Momentum and Angular Momentum | Physics Class 11 - NEETmwd2. The same is the angular momentum L2 of the second ball. The angular momentum of the system is equal to the sum of these two angular momenta i.e., L = 1/2 mwd2.

Conservation of Angular Momentum 

  • The time rate of change of angular momentum of a particle about some reference point in an inertial frame of reference is equal to the net torques acting on it.
  • Now, suppose that, Linear Momentum and Angular Momentum | Physics Class 11 - NEETthen, Linear Momentum and Angular Momentum | Physics Class 11 - NEET so that  Linear Momentum and Angular Momentum | Physics Class 11 - NEET= constant.
  • When the resultant external torque acting on a system is zero, the total vector angular momentum of the system remains constant. This is the principle of the conservation of angular momentum.
  • For a rigid body rotating about an axis (the z-axis, say) that is fixed in an inertial reference frame, we have L z = I w
  • It is possible for the moment of inertia I of a rotating body to change by rearrangement of its parts. If no net external torque acts, then L z must remain constant and if I do change, there must be a compensating change in w. The principle of conservation of angular momentum in this case is expressed. Iw = constant

Question for Linear Momentum and Angular Momentum
Try yourself:
A particle of mass m is moving in a circular path of radius r with a constant speed v. What can be said about the angular momentum of the particle about the center of the circle?
View Solution

Q4. A wheel of the moment of inertia I and radius R is rotating about its axis at an angular speed of w0. It picks up a stationary particle of mass m at its edge. Find the new angular speed of the wheel. 

Sol: The net external torque on the system is zero. Therefore, angular momentum will remain conserved. Thus,

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Note: 

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Comments on Linear Momentum : 

 Case I: Linear momentum is not conserved just before and just after collision because during collision hinge force acts as an external force.

Case II: Linear momentum is conserved just before and just after collision because no external force on the string.

Comments on Angular Momentum : 

Case I: Hinge force acts at an external force during collision but except point A all the other reference points given Linear Momentum and Angular Momentum | Physics Class 11 - NEET. So angular momentum is conserved only for point A.

Case II: Angular momentum is conserved at all points in the world.

Q5. A uniform rod of mass m and length l can rotate freely on a smooth horizontal plane about a vertical axis hinged at point H. A point mass having the same mass m coming with an initial speed u perpendicular to the rod, strikes the rod in-elastically at its free end. Find out the angular velocity of the rod just after collision.

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Sol: 

Angular momentum is conserved about H because no external force is present in the horizontal plane which is producing torque about H.

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Q6. A uniform rod of mass m and length l can rotate freely on a smooth horizontal plane about a vertical axis hinged at point H. A point mass having the same mass m coming with an initial speed u perpendicular to the rod, strikes the rod and sticks to it at a distance of 3l/4 from the hinge point. Find out the angular velocity of the rod just after collision.

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Sol:

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

From angular momentum conservation about H, initial angular momentum = final angular momentum

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Q7. A uniform rod AB of mass m and length 5a is free to rotate on a smooth horizontal table about a pivot through P, a point on AB such that AP = a. A particle of mass 2m moving on the table strikes AB perpendicularly at the point 2a from P with speed v, the rod being at rest. If the coefficient of restitution between them is Linear Momentum and Angular Momentum | Physics Class 11 - NEET, find their speeds immediately after impact. 

Sol: 

Let the point of impact be Q so that

PQ = 2a

Let P be the point of pivot that AP = a

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Let the velocities of point, Q, and the particle after impact be vq and vp respectively then from momentum conservation about point P.

Li = Lf 

Equilibrium of a Rigid Body

We can say rigid body is in equilibrium when it is in

(a) Translational equilibrium

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Fnet x = 0 and Fnet y = 0 and

(b) Rotational equilibrium

Linear Momentum and Angular Momentum | Physics Class 11 - NEET 

Note :

(i) If net force on the body is zero then net torque of the forces may or may not be zero.

Example:

(1) A pair of forces each of same magnitude and acting in opposite direction on the rod.

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

(2) If net force on the body is zero then torque of the forces about each and every point is same

t about B Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

t about C Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Translatory and Rotatory Equilibrium

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Q8. Determine the point of application of third force for which body is in equilibrium when forces of 20 N & 30 N are acting on the rod as shown in figure 

Linear Momentum and Angular Momentum | Physics Class 11 - NEET 

Sol: Let the magnitude of third force is F, is applied in upward direction then the body is in the equilibrium when

(i) Linear Momentum and Angular Momentum | Physics Class 11 - NEET (Translational Equillibrium)

⇒ 20 + F = 30 ⇒ F = 10 N

So the body is in translational equilibrium when 10 N force act on it in upward direction.

(ii)Let us assume that this 10 N force act. Then keep the body in rotational equilibrium So Tor que about C = 0

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

30 x 20 = 10 x 
x = 60 cm

Linear Momentum and Angular Momentum | Physics Class 11 - NEET 

so 10 N force is applied at 70 cm from point A to keep the body in equilibrium.

Q9. Determine the point of application of force, when forces are acting on the rod as shown in figure.

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Sol: Since the body is in equillibrium so we conclude Linear Momentum and Angular Momentum | Physics Class 11 - NEET and torque about any point is zero i.e.,

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Let us assume that we apply F force downward at A angle q from the horizontal, at x distance from B

Linear Momentum and Angular Momentum | Physics Class 11 - NEET
⇒ Fnet x = 0 which gives
F2 = 8 N
From Fnet y = 0 ⇒ 5 + 6 = F+ 3
⇒ F1 = 8 N
If body is in equilibrium then torque about point B is zero,
⇒ 3 x 5 + F1 x - 5 x 10 = 0
⇒    15 + 8x - 50 = 0
Linear Momentum and Angular Momentum | Physics Class 11 - NEET 

Q10. A uniform rod length l, mass m is hung from two strings of equal length from a ceiling as shown in figure. Determine the tensions in the strings ? 

Linear Momentum and Angular Momentum | Physics Class 11 - NEET 

Sol: Let us assume that tension in left and right string is TA and TB respectively. Then

Rod is in equilibrium then  Linear Momentum and Angular Momentum | Physics Class 11 - NEET

From Linear Momentum and Angular Momentum | Physics Class 11 - NEET
mg = TA + TB ...(1)
Linear Momentum and Angular Momentum | Physics Class 11 - NEET
Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Linear Momentum and Angular Momentum | Physics Class 11 - NEET 

Q11. A stationary uniform rod of mass `m', length `l' leans against a smooth vertical wall making an angle q with rough horizontal floor. Find the normal force & frictional force that is exerted by the floor on the rod? 

Linear Momentum and Angular Momentum | Physics Class 11 - NEET 

 Sol:  As the rod is stationary so the linear acceleration and angular acceleration of rod is zero.

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

Question for Linear Momentum and Angular Momentum
Try yourself:
A uniform rod of mass m and length l is rotating freely on a smooth horizontal plane about a vertical axis hinged at point H. A point mass of mass 2m moving on the table strikes the rod perpendicularly at a distance of 3l/4 from the hinge point. What is the angular velocity of the rod just after the collision?
View Solution

Principle of moments

Consider a light rod of negligible mass which is pivoted at a point along its length. Let two parallel forces F1 and Fact at the two ends at distances d1 and d2 from the point of pivot and the normal reaction force N at the point of pivot as shown in figure. If the rod has to remain stationary in horizontal position, it should be in translational and rotational equilibrium. Then, both the net force and net torque must be zero.

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

For net force to be zero, − F1 + N − F2 = 0

N = F1 + F2

For net torque to be zero, d1F1 − dF2 = 0

d1F= d2F2

The above equation represents the principle of moments. This forms the principle for beam balance used for weighing goods with the condition d1 = d2; F1 = F2. We can rewrite above equation  as,

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

If F1 is the load and F2 is our effort, we get advantage when, d1< d2. This implies that F1> F2. Hence, we could lift a large load with small effort. The ratio (d2/d1) is called mechanical advantage of the simple lever. The pivoted point is called fulcrum.

Linear Momentum and Angular Momentum | Physics Class 11 - NEET

There are many simple machines that work on the above mentioned principle.

The document Linear Momentum and Angular Momentum | Physics Class 11 - NEET is a part of the NEET Course Physics Class 11.
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FAQs on Linear Momentum and Angular Momentum - Physics Class 11 - NEET

1. What is the definition of linear momentum and how is it calculated?
Ans.Linear momentum is defined as the product of the mass and velocity of an object. It is a vector quantity, represented mathematically as \( \vec{p} = m \vec{v} \), where \( \vec{p} \) is the linear momentum, \( m \) is the mass, and \( \vec{v} \) is the velocity of the object.
2. How is angular momentum of a particle about a point determined?
Ans.The angular momentum \( \vec{L} \) of a particle about a point is calculated using the formula \( \vec{L} = \vec{r} \times \vec{p} \), where \( \vec{r} \) is the position vector from the point to the particle and \( \vec{p} \) is the linear momentum of the particle. The cross product indicates that angular momentum is a vector quantity that depends on both the position and linear momentum.
3. What is the relationship between torque and angular momentum?
Ans.The relationship between torque \( \vec{\tau} \) and angular momentum \( \vec{L} \) is given by the equation \( \vec{\tau} = \frac{d\vec{L}}{dt} \). This means that the torque applied to an object is equal to the rate of change of its angular momentum. If the net torque on an object is zero, the angular momentum remains constant.
4. How do you calculate the angular momentum of a rigid body rotating about a fixed axis?
Ans.The angular momentum \( \vec{L} \) of a rigid body rotating about a fixed axis can be calculated using the formula \( \vec{L} = I \vec{\omega} \), where \( I \) is the moment of inertia of the body about the axis of rotation and \( \vec{\omega} \) is the angular velocity. This relationship shows that angular momentum depends on both the distribution of mass and the speed of rotation.
5. What is the principle of conservation of angular momentum?
Ans.The principle of conservation of angular momentum states that in the absence of external torques, the total angular momentum of a system remains constant. This means that if no external forces act on a system, the angular momentum before an event (such as a collision) will equal the angular momentum after the event, allowing for the analysis of rotational motion in isolated systems.
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