1. Force (F):

Force is an external agent capable to change the state of rest, motion or direction of an object. The effect of force on the state of motion of body was first understood by Isaac Newton (1642 – 1727). Force is a vector quantity and thus has not only magnitude but also direction. So, if two or more forces act on a body, we find the net force (or resultant force) by adding them as vectors.

(i) Units:

(a) SI unit: newton $(N)$ or $kg m s^{- 2}$

(b) CGS unit: dyne $(10^{- 5} N)$

(d) Other units: poundal, pound-force, kip, kilo pond

(ii) Definition of one newton: The force required to produce an acceleration of $1 m s^{- 2}$ on an object of mass $1 kg$ .

(iii) Dimensions: ${MLT}^{- 2}$

(iv) Effect of resultant force:

(a) may change only speed.
(b) may change only direction of motion.
(c) may change both the speed and direction of motion.
(d) may change size and shape of a body.

2. Aristotle’s Fallacy:

An external force is required to keep a body in motion. The opposing forces such as friction (solids) and viscous forces (for fluids) are always present in the natural world. This explains why forces by external agencies are necessary to overcome the frictional forces to keep bodies in uniform motion.

3. Fundamental Forces in Nature:

(i) Gravitational force:

The force of interaction that exists between two particles of masses $m_{1}$ and $m_{2}$ , due to their masses is called gravitational force.

Gravitational force on mass $m_{1}$ due to mass $m_{2}$ is ${\vec{F}}_{12} = \frac{G m_{1} m_{2}}{{|{\vec{r}}_{21}|}^{3}} {\vec{r}}_{21} = \frac{G m_{1} m_{2}}{{|{\vec{r}}_{2} - {\vec{r}}_{1}|}^{3}} ({\vec{r}}_{2} - {\vec{r}}_{1})$

where, ${\vec{F}}_{12}$ is the force on mass $m_{1}$ due to mass $m_{2}$ and ${\vec{r}}_{21}$ is the position vector pointing from mass $m_{1}$ to mass $m_{2}$

(a) It is the weakest force and is always attractive.

(b) It is a long range force as it acts between any two particles situated at any distance in the universe.

(ii) Electromagnetic force:

Force exerted by one particle on the other because of the electric charge on the particles is called electromagnetic force.

(a) It can be attractive or repulsive.

(b) It is a long range force.

(d) All macroscopic forces (except gravitational) which we experience as push or pull or by contact are electromagnetic, i.e., tension in a rope, the force of friction, normal reaction, muscular force and force experienced by a deformed spring are electromagnetic forces. These are manifestations of the electromagnetic attractions and repulsions between atoms/molecules.

(iii) Strong nuclear force:

(a) It is the strongest force.

(b) It keeps nucleons (neutrons and protons) together inside the nucleus in spite of large electric repulsion between protons.

(c) Radioactivity, fission, and fusion, etc. result because of the unbalancing of nuclear forces. It acts within the nucleus that too up to a very small distance.

(iv) Weak force:

(a) It acts between any two elementary particles.

(b) Under its action a neutron can change into a proton emitting an electron and a particle called antineutrino.

4. Classification of Forces on the Basis of Contact:

(i) Field force:

Force that acts on an object at a distance due to the interaction of an object with the field produced by other object is called field force. E.g. Gravitational force, Electromagnetic force.

(ii) Contact force:

Forces which are transmitted between bodies by short range atomic molecular interactions are called contact force. When two objects come in contact they exert contact forces on each other. E.g. Frictional force, Normal reaction, Tension.

5. Newtonian Mechanics:

The study of impact of force on the motion of a body as Newton presented it, is called Newtonian mechanics. Newtonian mechanics does not apply to all situations. If the speed of the interacting bodies is very large, an appreciable fraction of the speed of light, we must replace Newtonian mechanics with Einstein’s special theory of relativity. If the interacting bodies are on the scale of atomic structure, we must replace Newtonian mechanics with quantum mechanics.

6. Galileo’s Law of Inertia:

Galileo studied motion of objects on an inclined plane. A ball released from rest on one of the planes rolls down and climbs up the other. If the planes are smooth, the final height of the ball is nearly the same as the initial height (a little less but never greater). In the ideal situation, when friction is absent, the final height of the ball is the same as its initial height. If the slope of the second plane is decreased and the experiment repeated, the ball will still reach the same height, but in doing so, it will travel a longer distance. In the limiting case, when the slope of the second plane is zero (i.e. is a horizontal) the ball travels an infinite distance. In other words, its motion never ceases. Galileo thus, arrived at a new insight on motion that had eluded Aristotle and those who followed him. The state of rest and the state of uniform linear motion (motion with constant velocity) are equivalent.

7. Newton’s First Law of Motion :

A body continues in its state of rest or uniform motion in a straight line unless compelled by an external unbalanced force to change that state. Newton’s laws of motion are valid only in a set of frame of references, these frames of reference are known as inertial frames of reference. Frames moving uniformly with respect to an inertial frame are themselves inertial. Generally, we take earth as an inertial frame of reference, but strictly speaking it is not an inertial frame.

(i) Mass:

The mass of a body is the characteristic of that body that relates the body’s acceleration to the net force causing the acceleration. Mass is scalar quantity and thus obeys the rules of ordinary arithmetic.

(a) Units:

SI Unit: $kilogram (kg)$

CGS Unit: $gram (g)$

(b) Mass is that property of a substance which specifies how much resistance an object exhibits to changes in its velocity i.e., it gives an idea of inertia of that body.

(c) Mass is an inherent property of an object and is independent of the object’s surroundings and of the method used to measure it. The mass of a closed system of bodies is independent of the processes going on in the system, no matter what kind these processes are.

(d) Mass should not be confused with weight. Mass and weight are two different quantities. Weight is the force with which Earth attracts the object and is dependent on mass.

(ii) Inertia:

Inertia is the property of the body due to which body opposes the change of the state. Inertia of a body is measured by mass of the body.

(a) Inherent property of all the bodies by virtue of which they cannot change their state of rest or uniform motion along a straight line by their own is called inertia.

(b) Inertia is not a physical quantity, it is only a property of the body which depends on mass of the body.

(d) Two bodies of equal mass, one in motion and another is at rest, possess same inertia because it is a factor of mass only and does not depend upon the velocity.

(e) Inertia of rest: It is the inability of a body to change by itself, its state of rest. This means a body at rest remains at rest and cannot start moving by its own.

(f) Inertia of motion: It is the inability of a body to change by itself its state of uniform motion i.e., a body in uniform motion can neither accelerate nor retard by its own.

(g) Inertia of direction: It is the inability of a body to change by itself it's direction of motion.

8. Newton's Second Law:

According to Newton’s second law of motion, the rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction in which the force acts.

(i) $\vec{F} = \frac{d \vec{p}}{d t} = \frac{d}{d t} (m \vec{v}) = m \frac{d \vec{v}}{d t} + \vec{v} \frac{d m}{d t}$ ( Linear momentum, $\vec{p} = m \vec{v}$ )

(ii) For constant mass system, $\vec{F} = m \vec{a}$

(iii) Force can’t change the momentum along a direction normal to it, i.e., the component of velocity normal to the force doesn’t change.

(iv) Newton’s second law is strictly applicable to a single point particle. In case of rigid bodies or system of particles or system of rigid bodies, $\vec{F}$ refers to total external force acting on system and $\vec{a}$ refers to acceleration of centre of mass of the system. The internal forces, if any, in the system are not to be included in $\vec{F}$ .

(v) Acceleration of a particle at any instant and at a particular location is determined by the force (net) acting on the particle at the same instant and at same location and is not in any way depending on the history of the motion of the particle

(vi) Momentum $(p)$ :

It is the product of the mass and velocity of a body i.e., momentum, $\vec{p} = m \vec{v}$ .

(a) It is a vector quantity and it’s direction is the same as the direction of velocity of the body.

(b) SI unit: $kg m s^{- 1}$

(d) Linear momentum of a body is the quantity of motion contained in the body.

(vii) Impulse $(I)$ :

Impulse is the product of force with time. Impulse of a force is a measure of total effect of force.

(a) For a finite interval of time from $t_{1}$ to $t_{2}$ then the impulse, $I = \int_{t_{1}}^{t_{2}} \vec{F} d t$

(b) If constant force acts for an interval $Δ t$ then Impulse, $\vec{I} = \vec{F} Δ t$

(d) Impulse is equal to the area under $F - t$ curve. If we plot a graph between force and time, the area under the curve and time axis gives the value of impulse.

(e) Impulse is a vector quantity and its direction is same as that of force.

(f) Units:

SI Unit: $kg m s^{- 1}$ or $N s$
CGS Unit: $g cm s^{- 1}$ or $dyne second$

(g) When a large force works on a body for very small time interval, it is called impulsive force. An impulsive force does not remain constant, but changes first from zero to maximum and then from maximum to zero. In such case, we measure the total effect of force.

9. Newton's Third Law of Motion:

Whenever a particle $A$ exerts a force on another particle $B$ , $B$ simultaneously exerts a force on $A$ with the same magnitude in the opposite direction.

${\vec{F}}_{AB} = - {\vec{F}}_{BA}$

(i) Action and reaction forces act on different bodies, not on the same body.

(ii) Magnitude of force is same on both the bodies.

(iii) Direction of forces are in opposite direction.

(iv) For action reaction pair, there is no need of contact.

(v) In an action reaction pair, both forces act simultaneously i.e., we can not say one as action and other reaction.

10. Common Forces in Mechanics:

(i) Normal force $(N)$ :

A contact force perpendicular (normal means perpendicular) to the contact surface that prevents two objects from passing through one another is called the normal contact force.

(a) It is the component of contact force perpendicular to the surface. It measures how strongly the surfaces in contact are pressed against each other. It is the electromagnetic force.

(b) Normal is a dependent force, it comes in role when one surface presses the other.

(d) Normal force acts along the common normal.

(e) Normal force acts perpendicular to the surface in contact.

(f) Normal force does not always equal $m g$ . If an object is on an incline, if there are applied forces with vertical components, or if there is a vertical acceleration of the system, then normal force is not equal to $m g$ .

(ii) Tension $(T)$ :

Tension in a string is a electromagnetic force. It arises when a string is pulled. If a massless string is not pulled, tension in it is zero. A string suspended by rigid support is pulled by a force ‘ $F$ ’ as shown in figure, for calculating the tension at point ‘ $A$ ’ we draw free body diagram $(FBD)$ of marked portion of the string. Here, string is massless. String is considered to be made of a number of small segments which attracts each other due to electromagnetic nature. The attraction force between two segments is equal and opposite due to Newton’s third law.

(a) Tension always acts along the string and in such a direction that it tries to reduce the length of string.

(b) If the string is massless then the tension will be same along the string but if the string have some mass then the tension will continuously change along the string.

(iii) Weight $(W)$ :

The weight W of a body is the magnitude of the net force required to prevent the body from falling freely, as measured by someone on the ground.

(a) Weight of an object is the force with which earth attracts it.

(b) It is also called the force of gravity or the gravitational force.

(c) Weighing machine: A weighing machine does not measure the weight but measures the force exerted by object on its upper surface.

(iv) Spring force:

For a spring that obeys Hooke’s law, the extension of the spring is proportional to the applied force. If we stretch spring by a distance x from its equilibrium position, it applies a restoring force ${\vec{F}}_{spring}$ , towards its equilbrium position.

${\vec{F}}_{spring} = - k \vec{x}$ where $x$ is displacement of the free end from its natural length or deformation of the spring and $k$ is spring constant.

(a) Spring property: $k l =$ constant, where $l =$ natural length of spring.

(b) If spring is cut into two in the ratio $m : n$ then spring constant is given by

$k l = k_{1} l_{1} = k_{2} l_{2}$

where, $l_{1} = \frac{m l}{m + n}$ and $l_{2} = \frac{n l}{m + n}$

Equivalent spring constant is given by

$\frac{1}{k_{eq}} = \frac{1}{k_{1}} + \frac{1}{k_{2}} + \dots$

(d) For parallel combination of springs:

Equivalent spring constant is given by

$k_{eq} = k_{1} + k_{2} + k_{3} + \dots$

(e) When spring is in its natural length, spring force is zero.

(f) Spring force doesn't change instantaneously, whereas, the tension in the string changes instantaneously. If a spring is cut, the tension in that spring becomes zero instantaneously

(g) Spring balance: It does not measure the weight. It measures the force exerted by the object at the hook.

11. Free Body Diagram (FBD):

(i) Two or more than two objects which interact with each other form a system. The system is what you choose to analyse motion.

(ii) Classification of forces on the basis of boundary of system:

(a) Internal forces: Forces acting each within a system among its constituents.

(b) External forces: Forces exerted on the constituents of a system by the outside surroundings are called as external forces.

(iii) Real force: Force that acts on an object due to other object is called as real force.

(iv) An isolated object (far away from all objects) does not experience any real force.

(v) A free body diagram $(FBD)$ consists of a diagrammatic representation of a single body or a subsystem of bodies isolated from its surroundings showing all the forces acting on it.

(vi) Steps for drawing $FBD$ :

(a) Identify the object or system and isolate it from other objects clearly specifying its boundary.

(b) First draw non-contact external force in the diagram. Generally, it is the weight.

(c) Draw contact forces that acts at the boundary of the object or system. Contact forces are normal, friction, tension and applied force.

(d) In $FBD$ , internal forces are not drawn, only external forces are drawn.

12. Law of Conservation of Linear Momentum:

If the net external force on a system is zero, then the momentum of the system is conserved.

(a) Law of conservation of linear momentum is independent of frame of reference, though linear momentum depends on frame of reference.

(b) Conservation of linear momentum is equivalent to Newton’s third law of motion.

13. Constrained Motion:

(i) String Constraint:

When two objects are connected through an inextensible tight string then the parameters of the motion of the object along the length of the string have a definite relation between them.

(a) Component of velocity along the string are equal, $v_{1} \cos (θ_{1}) = v_{2} \cos (θ_{2})$

(b) Relation between the velocity of pulley and the blocks using string constrain

$v_{P} = \frac{v_{1} + v_{2}}{2}$

(ii) Wedge Constraint:

Components of velocity along perpendicular direction to the contact plane of the two objects are always equal if there are no deformations and they remain in contact.

$v_{1} \sin (θ) = v_{2} \cos (θ)$

14. Newton’s Law for a System:

${\vec{F}}_{ext} = m_{1} {\vec{a}}_{1} + m_{2} {\vec{a}}_{2} + m_{3} {\vec{a}}_{3} + \dots$ where, ${\vec{F}}_{ext}$ is the net external force on the system, $m_{1}, m_{2}, m_{3}$ are the masses of the objects of the system and ${\vec{a}}_{1}, {\vec{a}}_{2} {\vec{a}}_{3}$ are the acceleration of the objects respectively.

(i) Equilibrium of a particle: Equilibrium of a particle in mechanics refers to the situation when the net external force on the particle is zero.

(a) Lami’s theorem: When three forces acting at a point are in equilibrium, then each force is proportional to the sine of the angle between the other two forces.

$\frac{A}{\sin (α)} = \frac{B}{\sin (β)} = \frac{C}{\sin (γ)}$

(ii) Problem Solving Strategy:

Newton’s laws refer to a particle and relate the forces acting on the particle to its mass and to its acceleration. But before writing any equation from Newton’s law, you should be careful about which particle you are considering. The laws are applicable to an extended body too which is nothing but collection of a large number of particles. Follow the steps given below in writing the equations :

(a) Step 1: Select the body
The first step is to decide the body on which the laws of motion are to be applied. The body may be a single particle, an extended body like a block, a combination of two blocks-one kept over another or connected by a string. The only condition is that all the parts of the body or system must have the same acceleration.

(b) Step 2: Identify the forces
Once the system is decided, list down all the force acting on the system due to all the objects in the environment such as inclined planes, strings, springs etc. However, any force applied by the system shouldn’t be included in the list. You should also be clear about the nature and direction of these forces.

(d) Step 3: Make a Free-body diagram (FBD)
Make a separate diagram representing the body by a point and draw vectors representing the forces acting on the body with this point as the common origin. This is called a free-body diagram of the body. Look at the adjoining free-body diagrams for the platform and the man. Note that the force applied by the man on the rope hasn’t been included in the FBD. Once you get enough practice, you’d be able to identify and draw forces in the main diagram itself instead of making a separate one.

(e) Step 4: Select axes and Write equations
When the body is in equilibrium then choose the axis in such a fashion that the maximum number of forces lie along the axis. If the body is moving with some acceleration then first find out the direction of real acceleration and choose the one of the axis along the real acceleration direction and other perpendicular to it. Write the equations according to the Newton’s second law in the corresponding axis.

(iii) Motion of a body on a smooth inclined plane:

(a) Natural acceleration of body down the plane, $a = g \sin (θ)$

(b) Force for acceleration $a$ up the plane, $F = m (a + g \sin (θ))$

(iv) Climbing on the Rope:

(a) Different cases:

Case 1: $T > m g \Rightarrow$ man accelerates in upward direction.

Case 2: $T < m g \Rightarrow$ man accelerates in downward direction.

Case 3: $T = m g \Rightarrow$ man’s acceleration is zero.

(b) Either climbing or descending on the rope man exerts force downward.

(v) Atwood Machine:

The ideal Atwood machine consists of two objects of mass $m_{1}$ and $m_{2}$ connected by an inextensible massless string over an ideal massless smooth pulley.

(a) Acceleration of blocks, $a = \frac{(m_{2} - m_{1}) g}{m_{1} + m_{2}}$

(b) Tension in the string, $T = \frac{2 m_{1} m_{2} g}{m_{1} + m_{2}}$

15. Newton’s Law for Non-Inertial Frame:

(i) Reference frame:

A frame of reference is basically a coordinate system in which motion of object is analyzed. There are two types of reference frames.

(a) Inertial reference frame: Frame of reference moving with constant velocity or stationary. Newton’s first law of motion is valid in inertial frame of reference.

(b) Non-inertial reference frame: A frame of reference moving with non-zero acceleration. Although earth is a non-inertial frame (due to rotation) but we always consider it as an inertial frame. A body moving in circular path with constant speed is a non intertial frame (direction change cause acceleration).

(ii) Pseudo force:

The force on a body due to acceleration of non-inertial frame is called fictitious or apparent or pseudo force and is given by $\vec{F} = - m {\vec{a}}_{0}$ , where, ${\vec{a}}_{0}$ is the acceleration of non-inertial frame with respect to an inertial frame and $m$ is mass of the particle or body. The direction of pseudo force must be opposite to the direction of acceleration of the non-inertial frame.

When we draw the free body diagram of a mass, with respect to an inertial frame of reference we apply only the real forces (forces which are actually acting on the mass). But when the free body diagram is drawn from a non inertial frame of reference a pseudo force (in addition to all real forces) has to be applied to make the equation $\vec{F} = m \vec{a}$ to be valid in this frame also.

${\vec{F}}_{real} + {\vec{F}}_{pseudo} = m \vec{a}$

Net sum of real and pseudo force is taken in the resultant force.

$\vec{a} =$ Acceleration of the particle in the non-inertial frame.

${\vec{F}}_{pseudo} = - m {\vec{a}}_{frame}$

(a) Direction of pseudo force is opposite to the acceleration of frame.

(b) Magnitude of pseudo force is equal to mass of the body which we are analyzing multiplied by acceleration of frame.

(iii) Apparent weight of a man in a lift:

(a) If the lift is moving with constant velocity $v$ upwards or downwards:

In this case, there is no accelerated motion. Hence, no pseudo force experienced by observer inside the lift.

Apparent weight, $W^{'} = M g = Actual weight$

(b) If the lift is accelerated upward with constant acceleration, $a$ :

Apparent weight, $W^{'} = M (g + a)$

Apparent weight, $W^{'} = M (g - a)$

(d) Special Case: If the acceleration of lift, $a = g$ :

Apparent weight, $W^{'} = 0$

Thus, in a freely falling lift the man will experience weightlessness.

(e) If lift accelerates downward with acceleration, $a > g$ :

Apparent weight, $W^{'} = M (g - a)$ is negative, i.e., the man will be accelerated upward and will stay at the ceiling of the lift.

16. Rocket Propulsion:

The initial momentum of the rocket on its launching pad is zero. When it is fired from the launching pad, the exhaust gases rush downward at a high speed and to conserve momentum, the rocket moves upwards.

(i) Thrust on the rocket: $F = - u \frac{d m}{d t} - m g$ where, $\frac{d m}{d t} =$ rate of change of mass of rocket = rate of fuel consumption = rate of ejection of the fuel and $u =$ velocity of exhaust gases. Here, negative sign indicates that direction of thrust is opposite to the direction of escaping gases. $F = - u \frac{d m}{d t}$ (if effect of gravity is neglected)

(ii) Acceleration of the rocket: $a = \frac{u}{m} \frac{d m}{d t} - g$

(iii) Instantaneous velocity of the rocket: $v = u \log_{e} (\frac{m_{0}}{m}) - g t$ , where $m =$ mass of rocket at any instant $t$ (instantaneous mass) and $m_{0} =$ initial mass of rocket

(iv) Burnt out speed of the rocket: $v_{b} = v_{max} = u \log_{e} (\frac{m_{0}}{m_{r}})$ , where $m_{r} =$ residual mass of empty container of the rocket.

The speed attained by the rocket when the complete fuel gets burnt is called burnt out speed of the rocket. It is the maximum speed acquired by the rocket.

17. Friction:

Friction is the force of two surfaces in contact, or the force of a medium acting on a moving object (i.e. air on aircraft). When two bodies are kept in contact, electromagnetic forces act between the charged particles (molecules) at the surfaces of the bodies. Thus, each body exerts a contact force on the other. The magnitudes of the contact forces acting on the two bodies are equal but their directions are opposite and therefore the contact forces obey Newton’s third law.

The direction of the contact force acting on a particular body is not necessarily perpendicular to the contact surface. We can resolve this contact force into two components, one perpendicular to the contact surface and the other parallel to it. The perpendicular component is called the normal contact force or normal force (N) and the parallel component is called friction (f). In some cases friction acts as a supporting force and in some cases it acts as an opposing force. Static friction and Kinetic friction are the two types of frictional force.

(i) Cause of friction:

Friction arises on account of strong atomic or molecular forces of attraction between the two surfaces at the point of actual contact. Friction is caused due to inter-locking of extended parts of one object into the extended parts of the other object.

(ii) Kinetic friction:

Kinetic friction exists between two contact surfaces only when there is relative motion between the two contact surfaces. It stops acting when relative motion between two surfaces ceases.

(a) Direction: It is opposite to the relative velocity of the object with respect to the other object in contact considered. Its direction is not opposite to the force applied it is opposite to the relative motion of the body considered which is in contact with the other surface.

(b) Magnitude: The magnitude of the kinetic friction is proportional to the normal force acting between the two bodies.

$f_{k} = μ_{k} N$

where, $N$ is the normal force and the proportionality constant $μ_{k}$ is called the coefficient of kinetic friction.

(c) The value of the coefficient of kinetic friction depends on the nature of the two surfaces in contact. If the surfaces are smooth $μ_{k}$ will be small, if the surfaces are rough $μ_{k}$ will be large. It also depends on the materials of the two bodies in contact.

(iii) Static friction:

It exists between the two surfaces when there is tendency of relative motion but no relative motion along the two contact surface.

The static friction force on an object is opposite to its impending motion relative to the surface. Steps to determine the direction of static friction force:

(a) Draw the free body diagram with respect to the other object on which it is kept.

(b) Include pseudo force also if contact surface is accelerating.

(d) The direction of static friction is opposite to the above component of resultant force.

The magnitude of static friction is equal and opposite to the external force exerted, till the object at which force is exerted is at rest. This means it is a variable and self adjusting force.

${\vec{f}}_{s} = - {\vec{F}}_{applied}$

It has a maximum value called limiting friction.

$f_{\max} = μ_{s} N$

where, $μ_{s}$ is the coefficient of static friction and $N$ is the normal reaction.

The actual force of static friction may be smaller than $μ_{s} N$ and its value depends on other forces acting on the body. The magnitude of frictional force is equal to that required to keep the body at relative rest.

$0 \leq f_{s} \leq f_{\max}$

(iv) Graph between applied force and force of friction:

(v) Coefficient of static friction $(μ_{s})$ and kinetic friction $(μ_{k})$ :

(a) They are dimensionless quantities independent of shape and area of contact.

(b) It is a property of the two contact surfaces.

(d) The value of friction coefficient can be from $0$ to $\infty$ .

(vi) Angle of friction $(λ)$ :

Angle of friction is the angle made by the resultant of normal reaction and limiting friction with the normal reaction.

(a) $\tan (λ) = \frac{f_{s}}{N} = \frac{μ_{s} N}{N} = μ_{s}$

(vii) Angle of repose $(θ)$ :

The maximum angle of an inclined plane for which a block remains stationary on the plane.

$\tan (θ) = μ_{s}$

For smooth surface, $θ = 0$

18. Dynamics of Circular Motion:

If there is no force acting on a body it will move in a straight line (with constant speed). Hence, if a body is moving in a circular path or any curved path, there must be some force acting on the body. In circular motion dynamics, we deal with the forces which are responsible for keeping an object in circular path.

(i) Centripetal force:

Force or combination of forces which provide centripetal acceleration necessary for the rotation of a particle is called centripetal force or radial force. Centripetal force is not a force itself. It is just the value of the net force acting along the inside normal which is responsible for circular motion. This force may be friction, normal, tension, spring force, gravitational force or a combination of them. For example:

(a) For an object tied to a string and is rotated on a smooth horizontal surface, tension is centripetal force.

(b) To revolve satellite around the Earth, gravitational force provides centripetal acceleration, so gravitational force is centripetal force.

(d) For an object placed on a rough rotating table, friction on the object due to table is centripetal force.

(ii) Problem solving strategy:

To solve any problem in uniform circular motion we identify all the forces acting along the normal (towards center), calculate their resultant and equate it to $\frac{m v^{2}}{r}$ . If the circular motion is non-uniform then in addition to above step we also identify all the forces acting along the tangent to the circular path, calculate their resultant and equate it to $m \frac{d v}{d t}$ .

(a) Identify the plane of circular motion.

(b) Locate the centre of rotation and calculate the radius.

(d) Resolve force along the radial direction and along the direction perpendicular to it.
(e) The net force along radial direction is mass times the radial acceleration i.e. $\frac{m v^{2}}{r}$ or $m ω^{2} r$ .

Note: Centripetal force is no separate force like Tension, Weight, Spring force, Normal reaction, Friction etc. In fact, any one of these or their combination may play a role of centripetal force.

(iii) Circular turning on a level road:

Suppose a car of mass $m$ is moving at a speed $v$ in a horizontal circular arc of radius $R$ . Three forces acting on the car are:

(a) Weight of the car, $m g$

(b) Normal reaction, $N$

The necessary centripetal force to the car will be provided by force of friction $f$ acting towards center, $f = \frac{m v^{2}}{R}$

Limiting value of friction, $f_{L} = μ N = μ m g (∵ N = m g)$

For safe turning without sliding,

$f \leq f_{L} \Rightarrow \frac{m v^{2}}{R} \leq μ m g \Rightarrow v \leq \sqrt{μ R g}$

If $μ$ and $R$ are known to us, the speed of the vehicle should not exceed $\sqrt{μ R g}$

If $v$ and $R$ are known to us, the coefficient of friction should be greater than $\frac{v^{2}}{R g}$

Similarly, on a rotating platform to prevent an object from sliding,

$f \leq f_{L} \Rightarrow m ω^{2} R \leq μ m g \Rightarrow ω \leq \sqrt{\frac{μ g}{R}}$

The maximum angular velocity, $ω_{\max} = \sqrt{\frac{μ g}{R}}$

(iv) Banking of road without friction:

When a vehicle is taking on horizontal circular turn, friction on tyres due to the road provides centripetal acceleration. Many times the friction is not sufficient enough, for example when speed is too large or the turn is too narrow. In such cases, instant of keeping the road horizontal, it is made tilted with the outer side as higher side. This is called banking of road and the angle with which the road is kept inclined is called angle of banking. This enables the normal force to provide a horizontal component to help friction to provide centripetal acceleration.

$N \sin (θ) = \frac{m v^{2}}{R} . . . (i)$

$N \cos (θ) = m g . . . (ii)$

Dividing eq. $(i)$ with eq. $(ii)$ , we get $\tan (θ) = \frac{v^{2}}{R g}$

Maximum safe speed, $v_{\max} = \sqrt{R g \tan (θ)}$

Note: The expression $\tan (θ) = \frac{v^{2}}{R g}$ also gives

(a) the angle of banking for an aircraft, i.e., the angle through which it should tilt while negotiating a curve, to avoid deviation from the circular path.

(b) the angle at which a cyclist should lean inward, when rounding a corner, where, $θ$ is the angle which the cyclist must make with the vertical.

(v) Banking of road with friction:

If a vehicle is moving on a circular road which is rough and banked also, then three forces may act on the vehicle, of these the first force, i.e., weight $(m g)$ is fixed both in magnitude and direction. The direction of second force, i.e., normal reaction $(N)$ is also fixed (perpendicular to road) while the direction of the third force i.e., friction $(f)$ can be either inwards or outwards while its magnitude can be varied upto a maximum limit $(f_{L} = μ N)$ . So, the magnitude of normal reaction $N$ and directions plus magnitude of friction $f$ are so adjusted that the resultant of the three forces mentioned above is $\frac{m v^{2}}{R}$ towards the center. Of these $m$ and $R$ are also constant. Therefore, magnitude of normal reaction and directions plus magnitude of friction mainly depends on the speed of the vehicle $v$ .

(a) Friction $f$ will be outwards if the vehicle is at rest $v = 0$ . Because in that case the component of weight $m g \sin (θ)$ is balanced by $f$ .

(b) Friction $f$ will be inwards if $v > \sqrt{R g \tan (θ)}$

(d) Friction $f$ will be zero if $v = \sqrt{R g \tan (θ)}$

(e) For maximum safe speed,

$N \sin (θ) + f \cos (θ) = \frac{m v^{2}}{R} . . . (i)$

$N \cos (θ) - f \sin (θ) = m g . . . (ii)$

As maximum value of friction, $f = μ N$

Putting the value of $f$ and then dividing eq. $(i)$ with eq. $(ii)$ , we get

$\frac{\sin (θ) + μ \cos (θ)}{\cos (θ) - μ \sin (θ)} = \frac{v^{2}}{R g}$

Maximum safe speed, $v_{\max} = {[\frac{R g (μ + \tan θ)}{(1 - μ \tan θ)}]}^{\frac{1}{2}}$

Similarly, minimum safe speed, $v_{\min} = {[\frac{R g (\tan θ - μ)}{(1 + μ \tan θ)}]}^{\frac{1}{2}}$

(vi) Normal reaction of road on a concave bridge:

(a) For concave bridge:

Normal reaction, $N = m g \cos (θ) + \frac{m v^{2}}{r}$

(b) For convex bridge:

Normal reaction, $N = m g \cos (θ) - \frac{m v^{2}}{r}$

(vii) Centrifugal force:

This pseudo force which has to be taken under account when the frame of reference is rotating object itself, since the rotating object is accelerated i.e. non-inertial frame of reference. The magnitude of this force is $\frac{m v^{2}}{R}$ or $m ω^{2} R$ (where, $R$ is radius of the circular path of the object) and its direction is always radially outwards.

When a body is rotating in a circular path and the centripetal force vanishes, the body would leave the circular path. To an observer $A$ who is not sharing the motion along the circular path, the body appears to fly off tangentially at the point of release. To another observer $B$ , who is sharing the motion along the circular path (i.e., the observer $B$ is also rotating with the body which is released, it appears to $B$ , as if it has been thrown off along the radius away from the centre by some force. This inertial force is called centrifugal force.)
Centrifugal force is a fictitious force which has to be applied as a concept only in a rotating frame of reference to apply Newton’s law of motion in that frame. $FBD$ of ball w.r.t. non inertial frame rotating with the ball.
Suppose we are working from a frame of reference that is rotating at a constant, angular velocity $ω$ with respect to an inertial frame. If we analyse the dynamics of a particle of mass $m$ kept at a distance $r$ from the axis of rotation, we have to assume that a force $m ω^{2} r$ react radially outward on the particle. Only then we can apply Newton’s laws of motion in the rotating frame. This radially outward pseudo force is called the centrifugal force.

(viii) Effect of Earth's rotation on apparent weight:

The earth rotates about its axis at an angular speed of one revolution per $24 hours$ . The line joining the north and the south poles is the axis of rotation. Every point on the earth moves in a circle. A point at equator moves in a circle of radius equal to the radius of the earth and the centre of the circle is same as the centre of the earth.

Apparent weight, $W_{app} = N = m g - m R ω^{2} \cos^{2} (θ),$ where $θ$ is latitude at the place.

(ix) Conical Pendulum:

A bob of mass $m$ attached to a light and in-extensible string rotates in a horizontal circle of radius $r$ with constant angular speed $ω$ about the vertical. The string makes angle $θ$ with vertical and appears tracing the surface of a cone.Hence, this arrangement is called conical pendulum.

$T \sin (θ) = m ω^{2} r . . . (i)$

$T \cos (θ) = m g . . . (ii)$

From eq. $(i)$ and eq. $(ii)$ , we get

(a) Angle of string with the vertical, $\tan (θ) = \frac{ω^{2} r}{g}$

(b) Angular velocity of the bob, $ω = \sqrt{\frac{g \tan (θ)}{r}} = \sqrt{\frac{g}{h}} = \sqrt{\frac{g}{L \cos (θ)}}$

(d) Time period, $T_{P} = \frac{2 π}{ω} = 2 π \sqrt{\frac{L \cos (θ)}{g}}$

(e) Tension in the string, $T = m g \sqrt{1 + {(\frac{v^{2}}{r g})}^{2}}$

Important Questions on Laws of Motion

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