Mechanics
The branch of physics which deals with the study of motion material object is called Mechanics.
Sub-branches of Mechanics
(i) Statics: The study of objects at rest or in equilibrium. The measurement of time is not essential in statics.
(ii) Kinematics: The study of motion of objects without considering the cause of motion. Here measurement of time is essential.
(iii) Dynamics: The study of motion of objects taking into consideration the cause of their motion.
Dynamics is concerned with the forces which cause motion. Newton contributed a lot to dynamics by postulating his famous laws of motion.
Rest
If a body does not change its position as time passes with respect to frame of reference, it is said to be at rest.
Motion
If a body changes its position as time passes with respect to frame of reference, it is said to be in motion.
Rest And Motion Are Relative Terms
A passenger sitting in a moving train is at rest with respect to his fellow passengers but he is in motion with respect to the objects outside the train. Thus an object may be at rest w.r.t. one object and at the same time it may be in motion relative to another object. Hence rest and motion are relative terms.
Absolute Rest And Motion Are Unknown
- No object in the universe is in a state of absolute rest.
- As no object in the universe is at absolute rest, so the absolute motion cannot be realised.
- Only relative rest and relative motion can be realised.
Types of Motion of a Body
- 1. Rectilinear or Translatory motion
- 2. Circular or Rotatory Motion
- 3. Oscillatory or Vibratory Motion
Rectilinear or Translatory motion
Rectilinear motion is that motion in which a particle or point mass body is moving along a straight line (motion in a straight Line).
Translatory motion is that motion in which a body, which is not a point mass body is moving such that all its constituent particles move simultaneously along parallel straight lines and all its constituent particles shift through equal distance in a given interval of time.
Circular or Rotatory Motion
A circular motion is that motion in which a particle or a point mass body is moving on a circle, about a fixed point which is the centre of a circle.
A rotatory motion is that motion in which a body, which is not a point mass body, is moving such that all its constituent particles move simultaneously along concentric circles, whose centres lie on a line, called axis of rotation and shift through equal angle in a given time.
Oscillatory or Vibratory Motion
Oscillatory motion is that motion in which a body moves to and fro or back and forth repeatedly about a fixed point (called mean position) in a definite interval of time.
In the oscillatory motion, the amplitude is very small, (i.e., microscopic), the motion of body is said to be a vibratory motion.
Particle
The smallest part of matter with zero dimension which can be described by its mass and position is defined as a particle.
Point object
If the position of an object changes by distances much greater than its own size in a reasonable duration of time, then the object may be regarded as a point object.
Examples:Earth can be regarded as a point object for studying its motion around the sun.
Frame of Reference
The frame of reference is a system of coordinate axes attached to an observer having a clock with him, with respect to which, the observer can describe position, displacement, acceleration etc. of a moving object.
Frames of reference can be of two types:
(a). Inertial frame of reference is one in which Newton’s first law of motion > holds good. For example, a frame of reference attached to a person in a bus at rest or in a bus moving with a uniform velocity along a straight path.
(b). Non-inertial frame of reference is one in which Newton’s first law of motion does not hold good. For example, a frame of reference attached to a person in a bus moving with variable velocity or moving with acceleration along a straight path.
Motion In One, Two And Three Dimensions
One dimensional motion – The motion of an object is said to be one dimensional if only one of the three coordinates specifying the position of the object changes with time.
Examples -Motion of a freely falling body.
Two dimensional motion -The motion of an object is said to be two dimensional if only two of the three coordinates specifying its position change with time.
Example: Motion of planets around the sun
Three dimensional motion –The motion of an object is said to be three dimensional if all the three coordinates specifying its position change with time.
Example: A kite flying on a windy day
Distance
Actual path length covered by a moving particle in a given interval of time is called distance.
- Distance is a scalar quantity.
- Dimension : [M0L1T0]
- Unit : metre (S.l.).
Displacement
The shortest distance between initial position and final position is called displacement.
- Displacement is a vector quantity
- Unit : metre (S.I.)
- Dimension : [M0L1T0]
Sample problems based on distance and displacement
Problem 1. A man goes 10m towards North, then 20m towards east then displacement is
[KCET (Med.) 1999; JIPMER 1999; AFMC 2003]
(a) 22.5m (b) 25m (c) 25.5m (d) 30m
Ans: 22.5m
Problem 2. A body moves over one fourth of a circular arc in a circle of radius r. The magnitude of distance travelled and displacement will be respectively
Ans: πr/2, r√2
Speed
Rate of distance covered with time is called speed.
- It is a scalar quantity having symbol.
- Dimension : [M⁰L¹T-1]
- Unit : metre/second (S.I.), cm/second (C.G.S.)
Types of Speed
Uniform speed: When a particle covers equal distances in equal intervals of time, (no matter how small the intervals are) then it is said to be moving with uniform speed.
Non-uniform speed : In non-uniform speed particle covers unequal distances in equal intervals of time.
Average speed : The average speed of a particle for a given ‘Interval of time’ is defined as the ratio of distance travelled to the time taken.
v={\lim_{{∆t}\rightarrow{0}}}\frac{∆x}{∆t}=\frac{dx}{dt}Note: 1 Averaged speed : When particle moves the first half of a distance at a speed of V1 and second half of the distance at speed V2 then average speed-
v_{av}=\frac{2v_{1}v_{2}}{v_{1}+v_{2}}
Note: 2 Average speed : When particle covers one-third distance at speed v1, next one third at speed v2 and last one third at speed v3, then average speed-
v_{av}=\frac{3v_{1}v_{2}v_{3}}{v_{1}v_{2}+v_{2}v_{3}+v_{3}v_{1}}
Instantaneous speed :The velocity of an object at a particular instant of time or at a particular point of its path is called its instantaneous velocity.
v={\lim_{{∆t}\rightarrow{0}}}\frac{∆x}{∆t}=\frac{dx}{dt}
Velocity
Rate of change of position i.e. rate of displacement with time is called velocity.
\overrightarrow{v}=\frac{displacement}{time}
- It is a scalar quantity.
- Dimension : [M0L1T–1]
- Unit : metre/second (S.I.).
Types of Velocity
Uniform velocity : A particle is said to have uniform velocity, if magnitudes as well as direction of its velocity remains same and this is possible only when the particles moves in same straight line without
reversing its direction.
Non-uniform velocity : A particle is said to have non-uniform velocity, if either of magnitude or direction of velocity changes (or both changes).
Average velocity : It is defined as the ratio of displacement to time taken by the body.
\overrightarrow{v}_{av}=\frac{displacement}{time}=\frac{∆x}{∆t}
Instantaneous velocity : Instantaneous velocity is defined as rate of change of position vector of particles with time at a certain instant of time.
v={\lim_{{∆t}\rightarrow{0}}}\frac{∆x}{∆t}=\frac{dx}{dt}
Sample problems based on speed and velocity
Problem. If a car covers 2/5th of the total distance with v1 speed and 3/5th distance with v2 then average speed is
Problem . A man walks on a straight road from his home to a market 2.5 km away with a speed of 5 km/h. Finding the market closed, he instantly turns and walks back home with a speed of 7.5 km/h. The average speed of the man over the interval of time 0 to 40 min. is equal to
Problem. A car moving on a straight road covers one third of the distance with 20 km/hr and the rest with 60 km/hr. The average speed is
Acceleration
The time rate of change of velocity of an object is called acceleration of the object.
- It is a vector quantity.
- Dimension : [M0L1T–2]
- Unit : metre/second²
- It’s direction is same as that of change in velocity (Not of the velocity)
- There are three possible ways by which change in velocity may occur
- When only direction of velocity changes
- When only magnitude of velocity changes
- When both magnitude and direction of velocity changes
Retardation
When the velocity of a body decrease with time, then acceleration becomes negative. Negative acceleration is also called deceleration or retardation.
Types of acceleration
Uniform acceleration : A body is said to have uniform acceleration if magnitude and direction of the acceleration remains constant during particle motion.
Note: If a particle is moving with uniform acceleration, this does not necessarily imply that particle is moving in straight line. e.g. Projectile motion.Non-uniform acceleration : A body is said to have non-uniform acceleration, if magnitude or direction or both, change during motion.
Average acceleration : The ratio of total change in velocity of the object to the total time interval taken.
a_{av}=\frac{v_{2}−v_{1}}{t_{2}−t_{1}}=\frac{∆v}{∆t}
Instantaneous acceleration : The acceleration of an object at a given instant of time or at a given point of its motion is called instantaneous acceleration.
a={\lim_{{∆t}\rightarrow{0}}}\frac{∆v}{∆t}=\frac{d^{2}x}{dt^{2}}
The acceleration at an instant is the slope of the tangent to the v – t curve at that instant.
Sample problems based on acceleration
Problem 1 : The displacement of a particle, moving in a straight line, is given by s= 2t² + 2t + 4 where s is in metres and t in seconds. The acceleration of the particle is
[CPMT 2001]
(a) 2 m/s² (b) 4 m/s² (c) 6 m/s² (d) 8 m/s²
Problem 2 : A body of mass 10 kg is moving with a constant velocity of 10 m/s. When a constant force acts for 4 sec on it, it moves with a velocity 2 m/sec in the opposite direction. The acceleration produced in it is
[MP PET 1997]
(a)3 m/s² (b) -3 m/s². (c) 0.3 m/s² (d) -0.3 m/s²
Problem 3 : A particle is moving eastwards with velocity of 5 m/s. In 10 sec the velocity changes to 5 m/s northwards. The average acceleration in this time is
[IIT-JEE 1982]
(a) Zero
(b) 1/√2 m/s² toward north-west
(c) 1/√2 m/s² toward north-east
(d) 1/2 m/s² toward north-west
Problem 4 : The displacement of the particle is given by y= a+by+ct²-dt⁴ .The initial velocity and acceleration are respectively
[CPMT 1999, 2003]
(a) b, -4d
(b) -b, 2c
(c) b, 2c
(d) 2c, -4d
Problem 5 : The displacement of a particle, moving in a straight line, is given by s = 2t²+2t+4 where s is in metres and t in seconds. The acceleration of the particle is
[CPMT 2001]
(a) 2 m/s²
(b) 4 m/s²
(c) 6 m/s²
(d) 8 m/s²
Problem 6 : A body starts from the origin and moves along the x-axis such that velocity at any instant is given by (4t²-2t) where t is in second and velocity is in m/s. What is the acceleration of the particle, when it is 2m from the origin?
(a) 28 m/s²
(b) 22 m/s²
(c) 12 m/s²
(d) 10 m/s²
Position – Time Graph
a. For an object at rest
An object is said to be stationary(rest) if its position does not change with time.
figure
So the position time graph of the stationary object is straight line parallel to time axis.
b. For an object in uniform motion
An object is said to be in uniform motion if it covers equal distances in equal interval of time.
Figure
- Here graph will be a straight line inclined of time axis.
- Slope of position-time graph gives velocity.
c. For object in variable velocity
- Various position – time graphs and their interpretation
- Sample problems based on position-time graph
Velocity – Time Graph
- Various velocity – time graphs and their interpretation
- Sample problems based on velocity-time graph
Equations of Kinematics
- When particle moves with zero acceleration
- When particle moves with constant acceleration
- Important points for uniformly accelerated motion
- Sample problems based on uniform acceleration
Motion of Body Under Gravity (Free Fall)
- If a body dropped from some height (initial velocity zero)
- If a body is projected vertically downward with some initial velocity
- If a body is projected vertically upward
- Sample problems based on motion under gravity
Motion With Variable Acceleration
- Sample problems based on variable acceleration