IEMDaily - Video Lecture Notes

Example

Constrained Motion

In a system of elements, free motion is restricted due to some constraints and any relative motion which arises due to these constraints is called constrained motion.
There are 3 types of constrained motion:

Completely constrained motion
Incompletely constrained motion
Successfully constrained motion

Example

Spring two mass system

Assume a spring of spring constant

k

, one end is attached to a block of mass

m_{1}

and other end is attached to block of mass

m_{2}

, they are placed on a smooth horizontal surface.

m_{1}

and

m_{2}

are moving with velocity

u_{1}

and

u_{2}

in the same direction respectively. (

u_{2} > u_{1}

)
After a certain time, at an instant velocity of both the blocks will be same and let it be

^{'} u^{'}

and take that extension of spring is

^{'} x^{'}

. As the floor is smooth we can use law of conservation of energy.

{m_{1} (u_{1})}^{2} + {m_{2} (u_{2})}^{2} = {(m_{1} + m_{2}) u}^{2} + {k x}^{2}

Definition

Frame of reference

Inertial frame of reference are frames of reference which have zero acceleration. Force on a body in an inertial frame of reference due to the reference frame is zero.

Non-Inertial frame of reference are frames of reference which have non-zero acceleration. Force on a body in an inertial frame of reference due to the reference frame is non-zero and opposite to the direction of acceleration of the frame of reference.

Definition

Pseudo Force

Pseudo Force also called as the fictitious Force, Inertial Force or d'Alembert Force is an apparent Force that acts on all masses whose Motion is described using a Non-Inertial frame of reference that is undergoing Acceleration with respect to an Inertial frame. (e.g. Rotating reference frame).

The direction of the Pseudo Force is always opposite to the direction of the acceleration of frame of reference. This force always acts on the body in motion which is under consideration. Since it is also a force, it depends on two parameters, the mass, and the acceleration.

Definition

Apparent weight of a man in a Lift/ Elevator

Suppose a person having mass

m

is standing on a weighing machine which is placed in a stationary lift.
In this case, the actual weight of a person is

m g .

This acts on that weighing machine which offers a normal reaction N, which is nothing but the reading of weighing machine.
This reaction exerted by on the man is the apparent weight of the man.
But this apparent weight will depend upon the motion of lift.

Let us see how it depends upon the motion of lift.
1. When the lift will be at rest or moving with a uniform velocity no matter upwards or downwards, the apparent weight,

N = m g

i.e., actual weight of a man.
From FBD,

N - m g = m (0)

N = m g

2. When the lift will be accelerating upwards with an acceleration

a

.From FBD we can see that

N - m g = m a

N = m (g + a)

In this case, the apparent weight of the man has become more than the actual weight.
3. When the lift will be accelerating downwards with an acceleration a. From FBD,

m g - N = m a

N = M (g - a)

In this case, the apparent weight of the man has become less than the actual weight.
Note : If at any moment of time, cable of lift get broke down, then the reading of weighing machine will be 0.
Because both man and lift will be freely falling in this case. This phenomenon is known as "weightlessness".

Definition

Definition of a System

A set of things working together as parts of a mechanism or an interconnecting network. In order to study motion for a particular set of objects system is defined and different physical quantities like momentum, energy etc are calculated.

Example

Total contact force between surfaces

Example: For the arrangements shown in figure, acceleration of block B is

\sqrt{3} m / s^{2}

upwards. Find the normal reaction (in

k N

) between the
surfaces of contact of the two blocks.

Solution: Let horizontal direction is

x

and vertical direction is

y

and

N

is normal force between inclined surfaces, perpendicular to surface
direction,then constrained motion is defined as

x = y c o t θ

then

a_{x} = a_{y} c o t θ

a_{x} = c o t 30 \times \sqrt{3}

a_{x} = 3 m s^{-} 2

F - N s i n θ = m a_{x}

5000 - N s i n 30 = 1000 \times 3

N = 4000 N = 4 k N

Example

Calculate acceleration and force on blocks kept in contact

Problem:
Two blocks are kept in contact with each other on a smooth surface. The force on the lighter block and its acceleration is:

Solution: Given :

F = 10

m_{1} = 6

m_{2} = 4

kg
Total mass of the system

M = m_{1} + m_{2} = 6 + 4 = 10

∴

Net acceleration of the system

a = \frac{F}{M} = \frac{10}{10} = 1

m / s^{2}

Thus acceleration of lighter block is

1

m / s^{2}

Also force acting on the lighter block

F = m_{2} a = 4 \times 1 = 4

N

Example

Readings of Weighing Machines

Two masses

m_{1} a n d m_{2}

which are connected with a light string, are placed over a frictionless pulley. This set up is placed over a weighing machine, as shown. Three combination of masses

m_{1} a n d m_{2}

are used, in first case

m_{1}

=

6 kg and

m_{2}

=

2 kg, in second case

m_{1}

=

5 kg and

m_{2}

=

3kg and in third case

m_{1}

=

4 kg and

m_{2}

=

4 kg. Masses are held stationary initially and then released. If the readings of the weighing machine after the release in three cases are

W_{1}, W_{2} a n d W_{3}

respectively then : Reading of the weighing machine

=

2T + weight of the machine. As weight of the machine is constant.

T = \frac{2 m_{1} m_{2}}{m_{1} + m_{2}}

So reading is maximum for the case

m_{1} m_{2}

is maximum as

m_{1} + m_{2}

in all cases is same.

Example

Motion of Objects with respect to frames accelerated in one dimension

For example, the magnitude of the pseudo force on a driver by the racing car he operates (mass of the driver is 80 kg), as it accelerates horizontally along a straight line from rest to

60

m/s in

8.0

s would be estimated by this:
There are two ways to solve this problem :
1. Using equations of motion to calculate acceleration and subsequently force :
initial velocity u