Document

1 COMPOSSED AND WRITTEN BY PROF. NAJEEB MUGHAL. GOVT. MUSLIM SCIENCE DEGREE COLLEGE HYD.

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Let’s G¤ To HOUSE OF SUCCESS COME FOR SUCCESS

CHAPTER 5

Torque ,equilibrium and angular momentum

Contents:

1.

CENTER OF MASS & CENTER OF GRAVITY.

2.

TORQUE &COUPLE

3.

EQUILIBRIUM AND CONDITIONS.

4.

ANGULAR MOMENTUM

5.

LAW OF CONSERVATION OF ANGULAR MOMENTUM

6.

EQUATIONS

7.

DIMENSIONS.

8. SHORT QUESTIONS AND ANSWERS

TECHNECHAL TERMR ELATIVE DEFINITIONS

Center of mass:

 Every physical system has associated with it a certain point whose motion characterizes the motion of the whole system. When the system moves under some external forces than this point moves as if the entire mass of the system is concentrated at this point and also the external force is applied at this point of translational motion. This point is called the center of mass of the system.

Explanation:

Center of mass of system of n point masses is that point about which moment of mass of the system is zero, it means that if about a particular origin the moment of mass of system of n point masses is zero, then that particular origin is the center of mass of the system.

The concept of center of mass is a pure mathematical concept. If there are n particles having mass m

1

, m

2

....m n

and are placed in space (x

1

, y

1

, z

1

), (x

2

, y

2

, z

2

) ..........(x n

, y n

, z n

) then center of mass of system is defined as (X c

, Y c

, Z c

) n  i m x i i n  i m y i i n  i m z i i

X = c m i

,

Y = c n  i m i and

Z = c n  i m i n  i

Where x i

, y i and z i are the coordinates of a particle of mass. X c

, Y c and Z c are the coordinates of center of mass.

Center of gravity:



 A point at which the force of gravity can be considered to act, such point is called “Center of Gravity”. The value of “g” is same through out the mass.

Explanation: n  i m g = i n  i m g = M g i

, where “M” is the Total mass of a body.

Hence, “M g” is known as total weight of a body. The center of gravity is lying at

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Let’s G¤ To HOUSE OF SUCCESS COME FOR SUCCESS the geometric center of bodies when are, sphere, or rectangular, or uniform cylinder and or rectangular.

Torque or Moment:



 The cross product of force and moment arm is called “Torque”.

Explanation:

Suppose a force “

F

” is applied on a particle at any point, with respect to point of axis rotation of position vector “ r

”, known as moment arm. (It is perpendicular distance from point of axis of rotation on to applied force). Torque is a vector quantity, denoted by “  ”.

The direction of torque is determined by “right hand rule”. The anti-clock wise torque is positive and Clock-wise is negative.

  

The magnitude of torque is given by,  = F r Sin  .

Torque is the tendency of a force.

Couple:



 When two equal and opposite forces act on a body, they produce a “couple”.

Explanation:

Suppose forces

F

and -

F

are acting on to points A and B on a rigid body. The magnitude of the resultant force is zero. So they can’t produce any motion, but their torque is not equal to zero.

The torque of force at A,=

F × OA

( anti clock wise about origin point )

The torque of force at B = -

F



× OB

(clock wise about origin point)

The torques are acting in anti clock wise direction, their resultant is,

F × OA

+ (-

F × OB

)

F



F



=

F

 

= couple



1



=

F

 

= couple

F d

= F d Sin  = couple

Hence, moment or torque of the couple is equal to the product of their force and arm of couple. The application of two and opposite forces applied tangentially to the wheel produce a couple.

DESCRIPTIVE PART



Equilibrium:



Equilibrium is a state of balance. When a body or a system is in equilibrium, there is no net tendency to change.

When a body is completely in the state of rest or moves with uniform motion then it is said to be in the state of “equilibrium”.

The body completely in the rest is called its “Static equilibrium”, and in the uniform motion is called its “Dynamic equilibrium”. There are two conditions of equilibrium

First condition of equilibrium: 

 “The vector sum of all horizontal forces acting on a body is equal to zero and the vector sum of all vertical forces is equal to zero. An object in equilibrium, the vector sums of all the forces acting it must be zero.”

Mathematical Derivation:

Suppose

F

1

,

F

2

, -------,

F n number of forces are acting on an object, according to this condition that,

The horizontal resultant forces that acting on an object is equal zero.

F + F + ----- + F n

= 0

F = F i + F j

,

1x 1Y

F = F i + F j

2x 2Y

,

- - - - -

F = F i + F j

F + F + ----- + F n



0








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F = F i + F j + F i + F j + - - - + F i + F

1x 1Y 2x 2Y nx nY



0

F = F i + + F i - - - + F i = 0 x Y



1x 2x

F = F j+ F j + - - - +F

= 0

1Y 2Y nY nx

 

1Y 2Y nY





0

F



0

Therefore,





F + F + - -

1x 2x nx



F =0 x

And, 

F + F + - - - - + F

1Y 2Y nY

F = 0 y

 

0



This is “first condition of equilibrium”.

In this condition, the body remains in translational equilibrium. A body either remains at rest or it moves with uniform motion, when no external force is acting. The linear acceleration is equal to zero. This is on the basis of first law motion.

Second condition of equilibrium: 

 “The vector sum of all the external torque’s acting on a body about any reference point is equal to zero”.


Suppose τ , τ , - - -

2

τ n

are acting on an object about axis, it will be in the rotational equilibrium.

τ = τ + τ +- - - - + τ n

2



τ = 0

The second condition can be explained, as “The body will be in rotational equilibrium, if the sum of external counter clock wise torques is equal to the external clock wise torques”



τ

Counter clock wise

=



τ clock wise

In this condition an object does not produce angular acceleration and it remains in the state of rest or moves with uniform angular motion. In this condition no external torques are acting and angular acceleration is equal to zero.

Angular momentum:



 T he angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the linear momentum of the particle by the cross product of the moment arm of the particle

, denoted by “

L

”.

Explanation:

Suppose a particle of mass “m” is at position “

r

”and linear momentum is “ p

” in inertial of reference with origin

“O”. The angular momentum is vector quantity; it is cross product of position vector “

r

” and linear momentum “ p

”.

L

= p



r

We know that, p

= m v

Therefore,

L

= m v



r

Its magnitude is given by, L = p r Sin 

The direction is determined by “right hand rule”. The unit of angular momentum is J-sec.

Law of conservation of Angular momentum:





The total angular momentum of system of particles conserved, if no external force acts on system.








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Suppose a system of particles is acting upon by external as well as internal force. We know that, the net force acting, on a particle of mass “m” is moving with velocity “ v

”. The rate of change of momentum is given by, d

F = p dt

  d r × F = r × p dt

 

τ = r × d dt

 

We know that, from the definition of angular momentum, l = r × p

By differentiating, with respect to time, d dt d dt

  d dt



r × p



   d



 dt 



 d dt p



 v

 d r d t

And p m v

Therefore, d l d t

= v × m v +

 d l d t

= 0 +  Because,

v × v = 0

If the no net external torque acting on the particle, Then,  d l



0 d t l

= constant

Thus the angular momentum of a particles is conserved, if net torque acting on it is zero. Angular momentum for number of particles is,

L = l + l + - - - + l n d d t

  d     d

L = l + l + - - - + l dt

1 d dt

2 dt

  d d t

 d L



0 d t

Thus, total angular momentum remains conserved.

Equations

1.



F x

= 0,



F

Y

= 0,



F z

= 0 and



F = 0.



= 0 2. L = m v r Sinθ = p r Sinθ 3.

p m v 4.

 

L t

5.

τ = r × d dt

 

6.

  

7. d

F = p dt

 

8.

L = p



r 9.

X = c n  i m x i i n  i m i

, Y = c n  i m y i i n  i m i and Z = c n  i m z i i n  i m i








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Dimensions

PHYSICAL QUANTITY & SYMBOL DIMENSION UNIT

Rotational kinetic energy K.E = = ½ m r

2  2

.

2

M L T

-2

Joule

Angular momentum L = p



r

2

M L T

-1

J.sec = Kg.m

-2

.sec

-1

.

Radius of gyration

Moment of inertia I=

 m r

2 2

M L kg.m

2

.

Torque or couple

 

L t

2

M L T



2

N.m

Short questions and Answers

Q. #1: A central force is one that is always directed towards the same point. Can a central force give to torque about that point?

Answer:

Central force can not produce a torque about the central point because the moment arm is zero.

Q. #2: Explain why a particle experiencing only one force can not be in equilibrium?

Answer

If a single force is acting on a particle then the condition of equilibrium is not applicable. It states there,



F=0 and



=0

Q. #3: Give an example of a body which is in motion , yet in equilibrium.

Answer

A paratrooper jumps from a helicopter, is the example of the question. When the parachute opens, the forces acts on paratrooper balance each other. Therefore he moves downward with a uniform velocity.

Q. #4: A block rests on a horizontal surface. What forces act on it?

Answer:

When a block rests on a horizontal surface then two forces are acting, one of them is the weight acting vertically up ward and other balancing force normal reaction acting vertically downward. A block rests on a horizontal surface, according to the first condition of equilibrium.



F x

= 0 Or W+ (- R) = 0

Q. #5: Is torque a vector or scalar quantity?

Answer: Yes, a torque is a vector quantity. It is the cross product of force and force arm.

  

Q. #6: I s it possible to calculate the torque acting on rigid body with out specifying the origin?

Answer:

No, it is not possible to calculate the torque acting on rigid body with out specifying the origin

Q. #7: Is torque independent of location of origin?

Answer:

Torque is not independent of location of origin.

Q. #8: Where does the center of gravity of rectangular block lie?

Answer:

The center of gravity of rectangular block lies at the center of block..

Q. #9:Give an example of a body which is in motion and yet in equilibrium.

Answer:

A paratrooper jumps from an airplane, is an example of equilibrium and yet in motion. Because, after jumping from an airplane, a paratrooper is under the influence of gravity and moving down ward with uniform velocity.

Q. #10: Define angular momentum and derive its units.








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Answer:

Angular momentum is the vector product of linear momentum and moment arm.

L = p



r

The unit of angular momentum is Joule-second .Angular momentum = p r

L = m v r

L = kg.m.sec

-1.

m.

L = kg.m

2

.sec

-1

.

L = kg. m

2

.sec

=1 sec.sec

-1

.

L = kg.m

2

.sec

-2

.sec.

L = J.sec. (Because, Joule = kg.m

2

.sec

-2

)

Q. #11: Define center of mass?

Answer: A point where total mass of a body is to be concentrated, such point is called “center of mass”.








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.

1. Center of mass: 

Statement:

A body rotates or there are several bodies that move relative to one another, their is one point that moves in the same path that a would under the same net force. That point is called the “Center of Mass” .

Explanation:

Center of Mass is a point at which an applied force produces acceleration but no rotation.

An example in our daily life, consider the motion of mass of the diver, the center of mass follows a parabolic path even when the diver rotates. This is the same parabolic path that a projected particle follows when acted on only by the force of gravity

(that is projectile motion).

English:

Pound-inch (lb-in)

1 lb-in = 0.113 Nm

Pound-foot (lb-ft)

1 lb-ft = 1.356 Nm

 








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Let’s G¤ To HOUSE OF SUCCESS COME FOR SUCCESS center of mass: The centre of mass is an imaginary point where one can assume the entire mass of the given system or object to be positioned

Momentum Conservation and Motion of Center of Mass: If the external forces acting on the system add up to zero, the centre of mass moves with constant velocity.

Angular Momentum: The angular momentum of a rigid object is defined as the product of the moment of inertia and the angular velocity.L = I x



Moment of inertia : The Area Moment Of Inertia of a beams cross-sectional area measures the beams ability to resist bending. The larger the Moment of Inertia the less the beam will bend. Moment of inertia, also called mass moment of inertia or the angular mass, ( SI units kg m

2

, British units slug ft

2

) polar of inertia : The Polar Area Moment Of Inertia of a beams cross-sectional area measures the beams ability to resist torsion. The larger the Polar Moment of Inertia the less the beam will twist. The larger the polar moment of inertia, the less the beam will twist, when subjected to a given torque. The SI unit for polar moment of inertia is m

4.

Torque: Force is the rate of transfer of angular momentum. A torque is an influence which tends to change the rotational motion of an object. One way to quantify a torque is, Torque = Force applied x force arm

The principle of moments: the sum of all the clock-wise moments about any point must have the same magnitude as the sum of all the anti-clockwise moments about the same point.

Comparison of Linear and Rotational Motion: When a rigid body such as a merry-go-round rotates around an axis, each particle in the body moves in its own circle around that axis. Since the body is rigid, all particles make one revolution in the same amount of time. i.e., they all have the same angular speed



.

Moment of Inertia: A rigid body rotating about a fixed axis AB, a particle 'p' of mass is rotating in a circle of radius 'r'.

Law of conservation of angular momentum: The total angular momentum of a system is constant in both magnitude and direction if the resultant external torque acting on the system is zero.

Stability and balance: If a body is displaced slightly, three different outcomes are possible.

1) The object returns to its original position, in which case it is said to be in Stable equilibrium;

2) The object moves even father from its original position, in which case it is said to be in unstable equilibrium;

3) And, The object remains in its new position, in which case it is said to be in Neutral equilibrium.

Rotational kinetic energy: A rigid object rotating about a z axis with angular speed



. The kinetic energy of the particle of mass m is ½ m v 2. The total kinetic energy of the object is called its rotational kinetic energy.

K.E. = ½ m r 2  2 .

Stress: stress is a quantity that is proportional to the force causing a deformation; more specifically, stress is the external force acting on an object per unit cross-sectional area.

Strain: strain is a measure of the degree of deformation. It is found that, for sufficiently small stresses, strain is proportional to stress; the constant of proportionality depends on the material being deformed and on the nature of the deformation. We call this proportionality constant the elastic modulus.

Elastic modulus: The elastic modulus is therefore the ratio of the stress to the resulting strain.

Young’s modulus: It is measures the resistance of a solid to a change in its length

Shear modulus: It is measures the resistance to motion of the planes of a solid sliding past each other

Bulk modulus: It is measures the resistance of solids or liquids to changes in their volume

States of equilibrium: If a body is displaced slightly, three different outcomes are possible.








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1) The object returns to its original position, in which case it is said to be in Stable equilibrium;

2) The object moves even father from its original position, in which case it is said to be in unstable equilibrium;

3) And, The object remains in its new position, in which case it is said to be in Neutral equilibrium.

Gyroscope : it is a device for measuring or maintaining orientation , based on the principles of angular momentum . The device is a spinning wheel or disk whose axle is free to take any orientation. This orientation changes much less in response to a given external torque than it would without the large angular momentum associated with the gyroscope's high rate of spin. Since external torque is minimized by mounting the device in gimbals , its orientation remains nearly fixed, regardless of any motion of the platform on which it is mounted.

 A rigid object is in equilibrium if and only if the resultant external force acting on it is zero and the resultant external torque on it is zero about any axis:



F = 0 and



= 0 .

The first condition is the condition for translational equilibrium, and the second is the condition for rotational equilibrium.  The force of gravity exerted on an object can be considered as acting at a single point called the center of gravity. The center of gravity of an object coincides with its center of mass if the object is in a uniform gravitational field.  Stress is a quantity proportional to the force producing a deformation; strain is a measure of the degree of deformation.  Strain is proportional to stress, and the constant of proportionality is the elastic modulus: Elastic modulus = stress

 Three common types strain of deformation are (1) the resistance of a solid to elongation under a load, characterized by Young’s modulus Y; (2) the resistance of a solid to the motion of internal planes sliding past each other, characterized by the shear modulus

S ; and (3) the resistance of a solid or fluid to a volume change, characterized by the bulk modulus B.  An object in equilibrium is at rest. It is not moving (or "translating") and it is not rotating.  An object at rest is in equilibrium or in static equilibrium. An object at rest is described by Newton's First Law of Motion. An object in static equilibrium has zero net force acting upon it.

 The First Condition of Equilibrium is that the vector sum of all the forces acting on a body vanishes



F = 0

 A rotational force, also known as a torque, depends upon the force and where that force is applied; torque = moment arm (force).

 An object in equilibrium does not move along a straight line -- it does not translate -- that means the sum of all the forces on it is zero. That was the first condition of equilibrium.

 An object in equilibrium also does not rotate. That means the sum of all the rotational forces on it is also zero. The sum of all the torques on an object is equilibrium is zero. This is the Second Condition of Equilibrium



= 0  The Center of Mass has been described as "a uniquely interesting point". It is interesting for motion as well as for equilibrium situations.

 When calculating the torque (or rotational force) caused by the weight of an object, the object may be treated as a rigid, mass less shell with all its mass or weight concentrated at the center of mass. For a regular, uniform object, this center of mass or center of gravity is located at the ordinary center of the object.

 If a small movement lowers the Center of Mass, the object is in unstable equilibrium. If a small movement raises the Center of Mass, the object us in stable equilibrium.

 We have seen that objects do not move when the sum of the forces acting on them is zero. We can use this fact to find the conditions for "static equilibrium": the condition an object is in when there are forces acting on it, but it is not moving. The conditions for static equilibrium are easy to state: the sum of the

(vector) forces must equal zero, and the sum of the torques must equal zero:



F = 0and



= 0.

 Newton's first law of motion says "A body maintains the current state of motion unless acted upon by an external force." The measure of the inertia in the linear motion is the mass of the system and its angular counterpart is the so-called moment of inertia. The moment of inertia of a body is not only related to its mass but also the distribution of the mass throughout the body. So two bodies of the same mass may possess different moments of inertia.

 The torque acting on the particle is proportional to its angular acceleration, and the proportionality constant is the moment of inertia. It is important to note that



=

 

is the rotational analog of Newton’s second law of motion, F = ma.








10 10


 The elastic limit of a substance is defined as the maximum stress that can be applied to the substance before it becomes permanently deformed.

 Gyroscope invented by Léon Foucault, and built by Dumoulin-Froment, 1852.

 The unit for torque is N.m. in S.I. system of units, is the same as that for energy. But the two quantities are different. The main difference is that energy is scalar, where as torque is a vector. The special name joule (1J = 1N.m. ) is used only for energy or work and never for torque.







Document

CHAPTER 5

Torque ,equilibrium and angular momentum

Contents:

TECHNECHAL TERMR ELATIVE DEFINITIONS

Center of mass:

Center of gravity:

Torque or Moment:

Couple:

DESCRIPTIVE PART

Equilibrium:

Angular momentum:

Law of conservation of Angular momentum:

Equations

Dimensions

Short questions and Answers

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Document

CHAPTER 5

Torque ,equilibrium and angular momentum

Contents:

TECHNECHAL TERMR ELATIVE DEFINITIONS

Center of mass:

Center of gravity:

Torque or Moment:

Couple:

DESCRIPTIVE PART

Equilibrium:

Angular momentum:

Law of conservation of Angular momentum:

Equations

Dimensions

Short questions and Answers

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