| Laws of Motion
- Features of Laws of Motion:
1. A particle at rest or moving with uniform velocity relative to an observer is said to be in the state of equilibrium.
2. If a particle is in equilibrium vector sum of the forces acting on it is zero [S = ]
3. If a particle at rest continues to be at rest or a particle moving with uniform velocity, continues to move with uniform velocity, the net force acting on the particle is zero. (Newton's 1st Law)
4. If a net force acts on a particle it accelerates.
5. The inability of a body to change its direction of motion by itself is inertia of direction. eg. The sparks coming out of a grinding stone are tangential to the rotating stone.
6. The linear momentum of a particle is the product of its mass and velocity.
7. According to Newton II law of motion, the rate of change of linear momentum of a particle is equal to the net force on the particle. The direction of the net force on the particle will be in the direction of the change in momentum of the particle and it need not be in the direction of velocity
Newton II law
8. If the mass of a particle is constant, the net force acting on the particle according to Newton II law is
9. The second law provides a method to measure force
1 kg wt = 9.8 Newton.
1 gm wt = 980 dyne
10. Forces acting for a short duration and which vary with time are impulsive forces.
Ex. : a) Collision between two billiard balls
b) Nail driven into a wall by striking it with a hammer.
11. The impulse of a force is defined as the product of the average force and the time interval for which it acts.
Impulse J = FAV D t =
Impulse momentum theorem =
- Inertial frames of reference :
a) Frames of reference in which Newton Laws of motion are applicable.
b) Inertial frames of reference move with uniform velocity relative to each other.
c) Force acting on an object due to its interaction with another object is called a real force.
d) All fundamental forces of nature are real.
e) Real forces form action, reaction pairs.
Ex. : Normal force, Tension, weight, spring force, muscular force etc.
f) Equation of motion relative to an observer in an inertial frame is
(m is the mass of the body having acceleration relative to the observer.
g) Observers in all inertial frames, measure the same acceleration for a given object but might measure different velocities.
h) Observers in all inertial frames, measure the same net force acting on a given object.
i) Basic laws of physics are identical in all inertial frames of reference.
j) Inertial frames of reference are called Newtonian or Galilean frames of reference.
- Non-Inertial frames :
a) Frames of reference in which Newton Laws are not applicable.
b) Such frames accelerate relative to inertial frames.
c) Earth by definition is a non-inertial frame, because of its rotation about its axis and revolution around the sun. However in many applications it can be approximated to an inertial frame because acceleration of the earth due to its rotation and revolution is negligible.
d) Examples for non-inertial systems :
1) Accelerating car
2) Car going round a curve
3) Merry go round
4) Artificial Satellite
5) Accelerating elevator etc.,
e) Pseudo Force : Force acting on an object relative to an observer in a non-inertial frame, without any interaction with any other object of the universe.
Pseudo force examples : (Centrifugal force, deflection of pendulum relative to accelerating car, gain or loss of weight experienced in an accelerating elevator.
g) Pseudo force does not possess a reaction counter part.
h) Pseudo force exist for observers only in non-inertial frames, such forces have no existance relative to an inertial frame.
i) In order to explain the dynamics of a body relative to non-inertial frame, we must consider real as well as Pseudo forces acting on the given body.
j) If is the acceleration of a non-inertial frame. The Pseudo force acting on an object of mass m, relative to an observer in the given non-inertial frame is
i.e. Pseudo force acts on an object opposite to the direction of acceleration of the non-inertial frame.
k) Equation of motion relative to non inertial frame is
where a1is the acceleration of body relative to non-inertial frame.
l) Centrifugal force : It is a pseudo force experienced radially outward by an object relative to the object, moving in a circular path relative to an inertial frame. The centrifugal force is given by .
(V = speed of object relative to inertial frame)
- ROCKET PROPULSION:
a) Rocket is an example of system of variable mass. Motion of the rocket is based on the principle of conservation of linear momentum and Newton's third law of motion.
b) Momentum of rocket before launching is zero. When exhaust gasses rushes out of the rocket with high velocity, to obey the law of conservation of linear momentum, the rocket rises upwards.
c) Instantaneous force on the rocket :
is the rate of mass of the exhaust gases coming out of the rocket and 'u' is the velocity of the exhaust gas at that instant of the time with respect to the rocket, then the instantaneous force on the rocket () is given by = --- . Here 'u' is negative with respect to the direction of motion of the rocket. In the absence of gravitational force and air resistance, the rocket raises under the action of this instantaneous force ().
d) If 'm' is the mass of the rocket and un-burnt fuel at any instant of time 't', then force with which the rocket moves against gravity ma = u - mg. Where 'a' is the instantaneous acceleration of the rocket.
e) Let the mass of rocket and fuel at instant t = 0 is m0and the initial velocity at that instant is V0. At an instant t = t let mass of the rocket and unburnt fuel is 'm' and velocity is V m = m0- .t
= (me+ mleft over fuel ) where meis the mass of empty rocket v = v0+ uloge() - gt
As time progresses, the mass of rocket decreases, the velocity of rocket increases or rocket accelerate.
f) Burnt out speed of the rocket : The speed of the rocket when the entire fuel gets burnt is called burnt out speed of the rocket which is the maximum speed attained by the rocket.
The burnt out speed vb= V0+ u loge- gt. where meis the mass of the empty rocket (i.e. mass of the rocket without fuel) :
1. The force of friction is a force of contact between two surfaces which acts tangentially to the two surfaces in contact, and opposes or tends to oppose, the relative motion between the two surfaces.
2. The force of friction is a self adjusting force and is macroscopically a non-conservative force. It is fundamentally an electromagnetic force.
3. (a) The force of contact between two surfaces, perpendicular to the surfaces in contact is called normal force (N).
(b) The value of normal force depends upon position of body.
(c) When body is placed in horizontal surface
N = mg
(d) When body placed in inclined surface
N = mg cos
(e) When body held against vertical surface
N = F
(f) Number of normal forces are number of surfaces in contact.
4. If f and N represent the force of friction and normal force between two surfaces. The net contact force is given by
5. The angle (q) made by the net contact force with the normal force is called angle of friction.
6. If q is the angle of friction
- Types of friction
7) Static Friction : The force of friction between two surfaces when the two surfaces are at rest relative to each other.
Characteristics of static friction :
a) When there is no component of external force parallel to the two surfaces in contact with each other, the force of static friction is zero.
b) When an external force is applied parallel to one of the surfaces in contact with each other, and the two surfaces are at rest relative to one another, the force of static friction between the two surfaces is equal to the applied force.
c) In the above case, if the surfaces continue to remain at rest relative to one another, the force of static friction always equals the external force and increases as the external force increases.
d) The force of static friction between two surface attains a maximum value when, the two surfaces just begin to slide or slip relative to each another.
e) The maximum force of static friction (fs) equals the external force applied parallel to one of the two surfaces, required to just cause sliding or slipping between the two surfaces.
f) The maximum force of static friction between two surface is also called limiting frictional force.
g) The limiting frictional force between two surfaces is given by ms is called the coefficient of static friction between the two surfaces and N is the normal force between the two surfaces.
h) The static frictional force between two surfaces is f msN
i) msis a dimensionless, unit less physical quantity which depends upon the nature of the surfaces in contact with each other, condition of the surfaces (presence of impurities, extent of roughness, presence of lubricant, temperature etc).
j) msbetween two given surfaces is independent of the normal force between the two surfaces.
k) ms> 0, it can also be greater than one, but in most of the cases it is less than one.
j) If qs is the angle of limiting friction between two surfaces tan qs = ms
- Kinetic friction or Sliding friction or Dynamic friction
a) It is the force of friction which acts between two surfaces, only when they slide or slip relative to each other.
b) The force of kinetic friction between two surfaces (fk) equals the force needed to be applied to one of the surfaces, parallel to the surfaces in contact with other, such that the two surfaces slide relative to each other with uniform velocity.
c) More force is needed to set a body into motion, than to keep it moving with uniform velocity on a surface.
d) The force of kinetic friction between two given surfaces is less than the limiting frictional force between the two surfaces (fk< fs)
e) The force of kinetic friction between two surfaces is independent of relative velocity between the two surfaces, provided the velocity is neither to low nor too large.
f) The force of kinetic friction between two surfaces is given by fk= mkN
g) mkis the coefficient of kinetic friction, which is a dimensionless, unitless constant and depends upon the nature of the surfaces in contact with each other and the condition of the surfaces in contact with each other.
f) mk< msfor a given pair of surfaces.
g) The angle of kinetic friction between two surfaces qkis given by tan qk= mk
h) qk< qkfor a given pair of surfaces
- Rolling friction :
a) Rolling friction comes into play when a body such as a wheel rolls on a surface.
b) Rolling friction arises out of the deformation of the two surfaces in contact with each other.
c) Greater the deformation, greater is the rolling frictional force.
d) The rolling frictional force is inversely proportional to the radius of the rolling body.
e) The rolling frictional force between two given surfaces is less than kinetic and limiting frictional forces.
f) If mRis the coefficient of rolling friction mR< mk< msfor a given pair of surfaces.
g) Ball bearings are used in machinery parts because rolling friction is least.
h) Radial tyres used in cars reduce rolling friction.
- Laws of friction : The force of friction between two dry surfaces is
a) Independent of area of contact of the two surfaces.
b) Directly proportional to the normal force between the two surfaces (f a N)
- Causes of friction:
a) Due to interlocking of the irregularities of the two surfaces
b) Due to cold welding of the peak points of the two surfaces
c) Due to surfaces adhesion i.e. due to intermolecular forces of attraction between the molecules of the two surfaces
- Methods to reduce friction
a) By lubricating the surfaces in contact with each other.
b) By polishing the surfaces in contact with each other.
c) Excess polishing increases friction rather than reducing it, because surface adhesion is increased.
d) Streamlining the shape of a body for minimizing air friction.
- Uses of friction in our daily life :
a) We walk on ground because of friction between our feet and ground.
b) Automobiles are driven because of friction between tyres and ground.
c) Nail driven into a wall is held in the wall because of friction
d) Friction helps us in holding our pen while writing.
e) Coal is transported across large distances on a conveyer belt because of friction.
f) Friction between the atmosphere and meteors causes them to burn out in the atmosphere before striking the earth.
- Disadvantages :
a) Most of the energy supplied to machines is wasted in doing work against friction. This reduces the efficiency.
b) Friction causes wear and tear of machinery parts.
- Block on a rough fixed horizontal surface
a) If applied force F = 0, the force of friction is zero.
b) If applied force F < fs, the block does not move and the force of friction is f = F
c) If applied force F = fs block just slides and frictional force fs = ms N
d) If we continue to apply a force F = fs, the block slides with an acceleration given by a = (ms- mk) g
e) Once the block slides, force of friction on the block is kinetic frictional force (fk)
f) If the block slides with an acceleration under the influence of an external force F, the acceleration of the block is
g) If the block slides with uniform velocity, the applied force is F = fk
- Sliding block on a horizontal surface coming to rest :
a) If a block having initial velocity u slides on a rough horizontal surface and comes to rest, the acceleration of the block is a = -mkg
b) Distance travelled by the block before coming to rest is
c) time taken by the block to come to rest is
- Pulling a block or roller
a) If the pulling force is such that F cos < fs, the block will be at rest and the force of friction between block and the surface is f = F cos
b) The normal force is N = mg - F sin
c) Force needed to just slide the body is where f is the angle of friction between the two surfaces.
d) If the applied force is greater than the above value, block slides with an acceleration and the force of friction between the block and the surface is fk.
e) The minimum possible force among all directions required to just move the body is mg sin or
where f is the angle of friction. The force must be applied at angle ? to the horizontal at an angle equal to angle of friction f.
- Pushing a block or Roller :
a) If the pushing force is such that F cos < fs, the block will be rest and the force of friction between the block and the surface is f = F cos .
b) The normal force is N = mg + F sin q
c) Force needed to just slide the body is where f is the angle of friction.
- A chain of uniform length 'L' is placed on a rough horizontal table. The coefficient of friction between the chain and table is m then the maximum fractional length of chain that can be hung freely from the edge of the table is =
Minimum fraction of length of chain that can be on the table is
Block pressed against a vertical wall :
A body of mass 'm' is pressed against a vertical wall with a horizontal force 'F'. The normal force is F. If the coefficient of static friction is ms, then block will be about to slide down if ms F = mg
- A vehicle is moving on a horizontal surface. A block of mass 'm' is stuck on the front part of the vehicle. The coefficient of friction between the truck and the block is 'm'. The minimum acceleration with which the truck should travel, so that the body may not slide down is a =
Block on a smooth inclined plane
a) N = mg cos
b) acceleration of sliding block(a = g sin )
c) If l is the length of the inclined plane and h is the height. The time taken to slide down starting from rest from the top is
d) Sliding block takes more time to reach the bottom than to fall freely from the top of the incline.
e) Velocity of the block at the bottom of the inclined plane is same as the speed attained if block falls freely from the top of the inclined plane.
f) If a block is projected up the plane with a velocity u, the acceleration of the block is a = -g sin
g) Distance travelled up the plane before its velocity becomes zero is
h) time of ascent is
- Block on a rough inclined plane
a) Angle of repose a : It is the angle of inclination of the inclined plane with the horizontal for which block just begins to slide down.
b) If is the angle of repose s= tan a
c) The angle of repose is the angle of static friction
d) The angle of inclination is q less than a, the block does not slide down, it is at rest. The force of friction f < fs and is equal to
f = mg sin [mg sin < fs]
e) If the angle of inclination is equal to . then the block is in limiting equilibrium. The force of friction is
f = fs = s mg cos [mg sin = fs]
f) If the inclination is maintained at , the block will eventually slide down with an acceleration equal to
g) If the block slides down the inclined plane with uniform velocity k= tan where is the angle of inclination of the inclined plane.
h) If , the block slides down with an acceleration given by a = g [sin - k cos ] [mg sin > fs]
i) If > , and the block slides down from the top of the inclined plane. Velocity at the bottom of the plane is
j) In the above case time of descent is
k) The time taken by a body to slide down on a rough inclined plane is 'n' times the time taken by it to slide down on a smooth inclined plane of same inclination and length, then coefficient of friction is
l) If a block is projected up a rough inclined plane, the acceleration of the block is a = -g[sin q + k cos ]
m) Force opposing the motion of the block is F = mg sin + kmg cos
n) The distance traveled by the block up the plane before the velocity becomes zero is
o) The time of ascent is In the above case the block will come down sliding only if q a.
p) In the above case if time of decent is n times the time of ascent, then
q) Force needed to be applied parallel to the plane to move the block up with constant velocity is
F = mg sin + kmg cos
r) Force needed to be applied parallel to the plane to move the block up with an acceleration a is
F = mg sin + kmg cos + ma
s) If block has a tendency to slide, the force to be applied on the block parallel and up the plane to prevent the block from sliding is
F = mg sin - smg cos