Physics Concepts for JEE Mains/ Advanced

Engineering
41
Physics

Rolling Motion of a body is a combination of both translational and rotational motion of a round shaped body placed on a surface. When a body is set in rolling motion, every particle of body has two velocities - one due to its rotational motion and the other due to its translational motion, and the resulting effect is the vector sum of both velocities at all particles.

Rolling Motion is classified in two categories - Pure Rolling & Rolling with Sliding.

Pure rolling is a case when the point of bottom contact of rolling body with ground remains at rest and body is considered to be rotating about this point of contact. For Detailed understanding of rolling motion see the video below - https://youtu.be/iMQBvpt-IoU

In pure rolling on a rough surface, the body experiences static friction if an external force is acting on it and as the point of contact remains at rest no energy is dissipated due to friction. There are many varieties of problems which are framed on the concept of pure rolling. To understand the applications of pure rolling see the example videos -

https://youtu.be/iMQBvpt-IoU

https://youtu.be/iW5TJ7EuqRc

https://youtu.be/HxE7TKMt8Eo

https://youtu.be/kH7CPAhNYTg

https://youtu.be/OCLjEUWx4zs

42
Physics

In case of pure rolling of a body, it moves in such a way that the point of contact of the body with the surface remains at rest and it happens when the velocity of the body relative to the surface is equal to the product of the body's radius and its angular velocity.

Because under this condition, the velocity of the point of contact of the body due to translational and rotation motion is equal and in opposite direction. If in case the two velocities of the point of contact of the body with the surface due to translational and rotational motion are not equal then the point of contact will no longer be at rest and will slip on the surface and will result sliding with rolling which is called 'Rolling Motion with Sliding'. There can be two cases of Rolling with Sliding - Forward Sliding and Backward Sliding.

To understand the cases as well as the analysis of motion of a body under rolling with sliding, see the video below:

https://youtu.be/oTOUEjUfqA4

In case of either Forward or Backward sliding, in the absence of external forces on the body, always due to the kinetic friction on the body it will move in such a way that after some time it will start pure rolling. Many problems are based on this logic.

To understand the applications of the same logic see the example videos given below:

https://youtu.be/MzhtZ4_RPxM

https://youtu.be/UM1O0VXFCWg

43
Physics

In dynamics we have studied that work is the amount of energy which is transferred in the process of displacement of a body by external force acting on it. The amount of work is calculated by the product of force magnitude and the displacement from the point of application of force in the direction of force.

Similarly we can calculate the work done by external torques on a body in rotational motion. This can be calculated by the product of torque magnitude and the angular displacement of the body in the direction of torque vector. Hence, work done in rotational motion can be positive or negative i.e. it can increase or decrease the kinetic energy of the body.

To understand the basics of work in rotational motion see the video below -

https://youtu.be/nkRZByCLxog

The work-energy theorem is also applicable here for the amount of work done on or by the body in rotational motion which increases or decreases its kinetic energy. The time rate of work done on a body in rotational motion is the power which can be given by the dot product of the torque and angular velocity of body in rotation.

To understand applications of work and power in rotational motion see the example videos below -

https://youtu.be/GhpQgT6a2U

https://youtu.be/zKCB7bGuQ3s

44
Physics

There are lot of variety of problems which forms the basis of understanding advance applications of the topic. Here we are discussing some advance illustration cases of rigid body dynamics on which many different types of problems are framed.

Static Equilibrium of Bodies: When a body is in static equilibrium then its center of mass remain at rest and orientation of body also remain fixed. The condition of a body to be in static equilibrium is that total vector sum of all forces acting on it as well as net vector sum of all torques acting on it must be zero. See the Illustration videos below to understand this:

https://youtu.be/_fUh5osTzCI

https://youtu.be/nwAX7JQiks0

https://youtu.be/DTnMF0CtXrQ

Energy & Momentum Conservation for Rotating Bodies: Momentum Conservation and Work-Energy theorem is applied in the rotating bodies in the same way like we do in translational motions. Only difference is here we take total kinetic energy which is sum of translational and rotational kinetic energies of the body. See the Illustration videos below to understand this:

https://youtu.be/WAQ0TVkytDU

https://youtu.be/oe_ltrEGyf8

https://youtu.be/5aPwwlpCAiI

45
Physics

There are several situations of collisions in rigid body dynamics on which different problems are framed. In such cases we need to conserve angular momentum of the system about a point with respect to which the net external torque on the body is zero. The most important thing in such cases is to locate a point in the given system about which angular momentum is to be conserved. With regular practice of different illustrations only this can be achieved. To understand the applications of concepts in such situations, watch the illustration videos given below:

https://youtu.be/Xg--twYgp9E

https://youtu.be/OHDjeGYdEBk

https://youtu.be/A98Xl_zj6xs

https://youtu.be/Kj6TEz423ws

https://youtu.be/9u1Ivxc1JnI

https://youtu.be/IX53Nd9EiyY

One very important application for rigid body dynamics is the study of a falling rod on a horizontal floor. There are many concepts are used in analyzing motion of this of rod in situation. To understand the same, watch the illustration videos given below:

https://youtu.be/iztZF0Nb-6E

https://youtu.be/6OnbbjwchaU

https://youtu.be/0igjzb5JSLE

46

Advance Applications of Rigid Body dynamics (Part-III) In applications of rotational motion one of the most important section is rolling motion and variety of problems are framed on rolling motion and its applications. Rolling motion problems are divided in two categories - Pure Rolling and Rolling with sliding. Majority of problems which are asked in pre engineering exams are based on pure rolling and its applications. Here for in-depth understanding of pure rolling and its applications, watch the illustration videos given below. https://youtu.be/NOHsiOF72OM https://youtu.be/oe_ltrEGyf8 https://youtu.be/HABl5tdLIWI https://youtu.be/g75K-7aNn1M https://youtu.be/qm6_mjhSiXM https://youtu.be/ni48H9Wivvs Although the cases of Rolling with Sliding are not very commonly asked in competitions but its important to understand the concept and its applications for overall development of mental aptitude for solving problems. For the same watch the illustration videos given below on the concept of rolling with sliding. https://youtu.be/6-N-LL0HhHc https://youtu.be/U8MJwrScKro https://youtu.be/tQ3kGMWhT5g

47

Advance Applications of Rigid Body dynamics (Part-IV)

Cases of Toppling: There are several problems framed on cases of toppling. A body topples over a point only if the line of action of its weight crosses this point in either direction where it can topple. Below is a case of toppling of a prism which is fundamental but very important in understanding the way on handling the cases of toppling. See the below illustration for this concept.

https://youtu.be/7nsdVlrPMgU

https://youtu.be/Kj6TEz423ws

Applications of Conservation Laws: There are several ways in which problems of rigid body dynamics are asked on the basis of Laws of Motion, Conservation of Momentum, Work-Energy theorem, Conservation of Angular Momentum, concept of Impulse and Angular Impulse. There are many cases in which more than one concepts are used in solving the given situation. See the illustration videos below which are based on different situations based on above concepts.

https://youtu.be/3FUpOqgAkFQ

https://youtu.be/sYajaDQxezc

https://youtu.be/OHDjeGYdEBk

https://youtu.be/qm6_mjhSiXM

Acceleration of points on a Rolling Body: Another important case in rotational motion is to analyze the acceleration of different points on a rolling body. When a body is in rolling then its different points can have both tangential and normal acceleration which are to be analyzed very carefully. To understand the same please see the illustration video given below.

https://youtu.be/t-zwXS1gIUU

48

Introduction to Gravitational Force & its vector form

Newton's Law of Gravitation is a fundamental law we study in early classes. This law gives the gravitational force of attraction between two bodies located at some separation. This law forms the basis of understanding gravitational field theory. To cover the basics related to Newton's Law of Gravitation, see the video below:

https://youtu.be/o7tOrmxxBYQ

There are some limitations of Newton's Law of Gravitation, the cases under which the law is not applied. To understand these limitations, see the video below:

https://youtu.be/Wrsb7v4UgFY

When on a body, two or more gravitational forces act, then for analysing the equilibrium of body in state of rest, or for analysing the motion of body if it is in motion, we need the vector sum of all forces acting on the body, so first we need to understand how to represent the gravitational force in vector form. See the video below for this:

https://youtu.be/1LlSN-g31mQ

49

Physics Concepts for JEE 2016: Principle of Superposition of forces & Gravitational Field

https://farm3.staticflickr.com/2918/14577441420_c3fe4a7436_c.jpg

When two or more gravitational forces act on a body then the resulting effect on this body is analysed by principle of superposition of forces which is stated as the effect is considered by resultant force on the body calculated by vector sum of all independent gravitational forces acting on the body. See the video below to understand the principle of superposition of gravitational forces:

https://youtu.be/OL5X4NOjEV0

For understanding the application of principle of superposition, see the example videos below:

https://youtu.be/Oyla6AfKOqw

https://youtu.be/aQliarVGFAU

https://youtu.be/LbpYfvO0mS4

Introduction to Gravitational Field: The region in surrounding of every mass is the gravitational field, which is defined as a region in which any particle or extended body having mass experiences a force on it. The magnitude of the force per unit mass is defined as the gravitational field strength at a specific point in space of gravitational field region. To understand the basics of gravitational field, see the video below:

https://youtu.be/9NdfRdafEnc

50

Physics Concepts for JEE 2016: Gravitational Field due to Extended Bodies

https://upload.wikimedia.org/wikipedia/commons/thumb/0/05/Libyan_pivot_irrigation_460142568_02e969004a_o.jpg/640px-Libyan_pivot_irrigation_460142568_02e969004a_o.jpg

As already covered, the gravitational field strength at a point in space is given by the force experienced by a unit mass placed at that point, and gravitational field strength due to a point mass can be easily calculated by the ratio of the force experienced by a test mass placed at that point.

In case of an extended body to calculate the gravitational field strength at any point in its surrounding, we consider an elemental mass in the body and find the gravitational field strength at that point due to this elemental mass, and integrate the result for the whole mass of body. To understand this, see the video given below.

https://youtu.be/44LgkILFOYI

Using the above logic, gravitational field strength due to various bodies can be calculated. There are several bodies considered as standard bodies of which gravitational field can be calculated using this logic, and are used in forming the basics of many fundamental numerical problems. See the videos given below for gravitational field strength due to some standard bodies.

Gravitational Field Strength due to a Uniform Rod: https://youtu.be/ab61X9iQtcg

Gravitational Field Strength due to a Uniform Ring: https://youtu.be/z42ypASf-Uk

Gravitational Field Strength due to a Circular Arc: https://youtu.be/7TsMKbqOo4Q

51
Physics

Gravitational Field due to a sphere can be analyzed in two cases of Hollow and a Solid Sphere. For a hollow sphere by symmetry we can analyze that net gravitational field strength at interior points is equal to zero and it can also be analyzed by considering two diametrically opposite elements on the surface of shell. For outer points of the shell by spherical symmetry it is similar to that of a point mass. See the below video for understanding of the same:

https://youtu.be/t0JxnLAFUU0

Gravitational Field due to a solid sphere can be analyzed by its spherical symmetry for outer points like a spherical shell and for interior points we can consider a point inside of the sphere and divide the sphere in two parts, one inner sphere of radius up to the point and one outer shell of inner radius up to the point and outer radius equal to that of the sphere. See the video below for the analysis:

https://youtu.be/_emeL_R1TmM

Gravitational Field due to a Cylinder at points in its outer surrounding the field direction will be radial and similar to that of a long linear mass distribution. For this analysis we find the gravitational field strength due to a long thread of uniform linear mass density. At outer points of cylinder the field will remain the same. See the video below for the analysis:

https://youtu.be/ycekUojt6rA

52
Physics

Null Point between Earth & Moon: This is the point on line joining the centres of Earth & Moon where the Gravitational Field is zero. See the example video below to understand finding the null point between two bodies:

https://youtu.be/er3RUaYRqvY

Gravitational Field due to Subtractive Bodies: When from a body one part is removed then the gravitational field at surrounding points decreases due to the mass removed from the body. The below example videos illustrates the gravitational field due to a sphere from which a spherical mass is removed and a cavity is left:

https://youtu.be/jZ_ljOZebYE

https://youtu.be/dIa8BeOXH_I

Circular Motion under Gravitational Force : When point mass revolves around another mass in uniform circular motion under its gravitational force, then the necessary centripetal force required for the circular motion is provided by the gravitational force only. See the example video on an illustration on uniform circular motion under the gravitational field of a long cylindrical mass:

https://youtu.be/DS-zOPG2S1g

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Sir could you please tell which are the cases where we can use the Virtual Work Theorem. I cannot find the article on constrained motions .

54
Physics

Gravitational Lines of forces are imaginary lines in the region of gravitational field which gives information about the net gravitational field in the region at any point. The information includes the direction of gravitational field vector as well as the idea about strength of the gravitational field strength.

At any point in a region of gravitational field the tangent to a gravitational line of force gives the direction of field vector at this point and the density of gravitational lines in the neighborhood of that point gives the idea about the strength of the field. If in a region lines are closely spaced or denser than in the close neighborhood of this point gravitational field strength is considered to be more compared to a region where line density is less.

See the below video for the detailed explanation -

https://youtu.be/G4r-ZnBPn7w

55
Physics

Earth being considered like a solid sphere, we can use the result of gravitational field strength of a solid sphere for obtaining the field strength due to Earth at various locations on surface of Earth, inside Earth, and at some altitude above the Earth surface.

Like the case of a solid sphere, outside the surface of Earth also gravitational field strength varies as inversely proportional to square of the distance from Earth center, we can also express the result as a function of the height above the Earth surface.

Similar to this inside the Earth gravitational field decreases with linear variation with distance from Earth center. See the below videos for the derivation of results of gravitational field strength at various points due to Earth.

Gravitational Field on Surface of Earth : https://youtu.be/3sMpBgwFykw

Gravitational Field at a height above the Surface of Earth : https://youtu.be/9fdATrvOhOs

Gravitational Field at a depth inside the Surface of Earth : https://youtu.be/alH5e0Q77mI

56

Physics Concepts for JEE 2016: Effect of Shape and Earth Rotation on gravitational field

http://fi.ge.pgstatic.net/articles/processed/1448024721.45a131b5fc804cc69e5e97a0c5604b38.jpg

We discussed that value of gravitational field strength at different locations on Earth Surface, at an altitude or at a depth are different. Similarly value of g is also affected by other factors like shape of Earth and rotational motion of Earth.

Effect on value of g due to shape of Earth: In analysis of calculation of gravitational field strength on surface we considered Earth as a solid sphere. But actually Earth shape is ellipsoidal not exactly spherical due to which the results of gravitational field strengths of a solid sphere are not appropriate to use. The shape of Earth is flat at poles and having more curvature at equator due to which gravitational field strength is slightly more at poles then equator. To see the detailed understanding watch the video here: https://youtu.be/QnIH92fgmbw

Effect on value of g due to rotational motion of Earth: Every object or a body placed on Earth surface is in circular motion with centre located on the axis of rotation of Earth. Thus due to the accelerated motion Earth surface is a non-inertial reference frame and for analysis of dynamics of any object on Earth surface we must use centrifugal force on the body in radial direction away from centre of circular motion of the body. Due to this inertia force on body the effective weight of body decreases.

Also at different locations on Earth surface the radius of circles in which the bodies are moving are also different so centrifugal force considered will also be different so variation in value of g will also be different. See the video here for detailed understanding: https://youtu.be/lNJ2icc1fPk

57

Physics Concepts for JEE 2016: Gravitational Interaction Energy of a System of Particles

https://farm1.staticflickr.com/755/23255016946_147110c19d_c.jpg

Interaction Energy of any system of particles is the work done in assembling the system from the zero energy state against the interaction forces of the system. In case of gravitational interaction energy of a system of particles we consider zero energy state when the particles are at very large separation or we consider at infinity.

To understand the Gravitational Interaction Energy, see the video below:

https://youtu.be/V84lHsa3J5Y

To understand the calculation of Gravitational Interaction Energy for a given system of particles, see the video below:

https://youtu.be/gXoXdtIqN0E

To study the applications of the concept in different physical situations, see the example videos below:

https://youtu.be/aZARKnjzQuE

https://youtu.be/RLV2Ab0_CFo

58

Physics Concepts for JEE 2016: Gravitational Potential and its relation with Gravitational Field

https://farm3.staticflickr.com/2838/11106539483_320911bee1_c.jpg

In the region of the Gravitational Field at any point in space, a characteristic property is defined, called the Gravitational Potential, which is defined as the gravitational interaction energy of a unit mass, placed at that point, or alternatively defined as the work done in bringing a unit mass from infinity to that point, against the gravitational forces acting on it.

If at any point in space, gravitational potential is V, then the gravitational interaction energy of a mass m placed at this point is directly given as mV.

As gravitational forces are attractive in nature, the gravitational potential is always a negative value. Using gravitational potential, we can also find the work done in displacing a mass from one point to another, by or against gravitational forces by using the relation

Work = mass x gravitational potential difference

To understand gravitational potential from basics see the video below -

https://youtu.be/2ZqN_GJOUyI

At any point in space, the force experienced by a unit mass gives the gravitational field strength at that point in space. If we wish to calculate the gravitational potential at any point, we need to calculate the work done in bringing the unit mass from infinity to this point, so it can be calculated by integrating the elemental work done in displacing a unit mass by an elemental distance dx, within limits from infinity to that given point. Using this analysis we can find the relation between g and V in space, which will have a similar relation that exists between force and potential energy in a conservative field. To understand the same see the video below -

https://youtu.be/4boyaWjpvWE

59

Physics Concepts for JEE 2016: Gravitational Potential due to a point charge

http://photo.foter.com/photos/48/foucaults-pendulum.jpg

In the surrounding of a point charge when a unit point mass is brought from infinity to any point work done is calculated by line integral of gravitational field within limits from infinity to that point. This work will be the gravitational interaction energy of a unit mass or the gravitational potential at that point. To understand the analysis see the video below

https://youtu.be/dFmzYKZT8J8

Work Done in Gravitational Field: If we know the gravitational potential in space at a point V then the work done in bringing a mass m from infinity to this point is given as mV which is also the gravitational interaction energy of the point mass at that point. So if in a region of space if a mass m is transported from one point to another the work done in this displacement can be directly calculated by finding the difference in gravitational potential energy. To understand this analysis see the video below -

https://youtu.be/8rFubUCGr2Y

60

Gravitational Potential due to a Sphere

A sphere with uniform mass distribution is a symmetrical body and in its surrounding the gravitational field can be considered in radially inward direction just like a point mass due to the symmetry of its mass distribution. We can discuss the Gravitational Potential due to uniform hollow and solid spheres.

Gravitational Potential due to a Hollow Sphere

: In surrounding region of a uniform hollow sphere the gravitational potential can be considered like a point mass and same expression of potential due to a point mass is valid for the gravitational potential outside a uniform hollow sphere including the points on its surface (for all x greater than or equal to its radius R).

At interior points as we know that the gravitational field is zero, no work is required in displacement of a point mass inside the hollow sphere so gravitational potential must be same everywhere inside the sphere which will be equal to that of the surface. For analysis and discussion on gravitational potential due to a uniform hollow sphere watch the video below -

https://youtu.be/dCuUhJLl_LU

Gravitational Potential due to a Solid Sphere : Similar to the case of a hollow sphere in outside region of a solid sphere also gravitational potential can be given by the expression of potential due to a point mass because of symmetry including the outer surface points of the sphere.

At interior points as gravitational field is non-zero we need to calculate the potential at interior points of the sphere by calculating the potential difference between surface points and any interior point. The analysis and discussion on gravitational potential due to a solid sphere is analyzed in the video given below -

https://youtu.be/rm3x2X0X_Sc