Force and Laws of Motion

NEWTON’S FIRST LAW OF MOTION

• • Newton’s first law of motion states that: A body at rest will remain at rest, and a body in motion will continue in motion in a straight line with a uniform speed, unless it is compelled by an external force to change its state of rest or of uniform motion.
  • A block kept on the floor will remain at rest until some external force is applied to it. Also, it takes more effort or forces to move a heavy mass. This is directly related to a property known as Inertia. So, Newton’s First law is also known as the law of inertia.

INERTIA AND MASS

• If we try to push two different objects, one having larger mass than the other, we find that larger force is required to change the state of the body with larger mass. When force is applied then the body tries to resists change in its state of uniform motion or rest.
• Inertia: The property of a body due to which it resists a change in its state of rest or of uniform motion is called inertia.
• The body with larger mass requires larger force. Hence we can say it has greater inertia.
• Mass: Mass is a measure of the inertia of a body. And so, heavier objects have more inertia than lighter objects.
• Inertia in rotational motion is analogous to mass in linear motion.
• Examples of Inertia:
  • When a hanging carpet is beaten with a stick, the dust particles start coming out of it.
  • When a car or bus starts suddenly, the passengers fall backward. Similarly, a person tends to fall forward when a bus stops suddenly.
  • A book lying on a table will remain there until an external force is applied on it to remove or displace it form that position.

INERTIA IN ROTATIONAL MOTION

Consider a rotating disc. The kinetic energy of the disc can be represented as

$K. E. = \frac{1}{2} \sum _{}^{} m_{i }{v}_I^2 .................\left(1\right)$

But in this case each mass does not have same linear velocity hence it is easier to realize using angular velocity. Angular velocity, is equal for all the masses. We also know,

Substituting the value ofin (1) we get,

From the above equation, we get,

is the moment of inertia of the system.

UNIT OF MOMENT OF INERTIA

Unit of moment of inertia is

MOMENT OF INERTIA

There are two theorems to calculate moment of inertia about any arbitrary axis:
o Parallel axis theorem and
o Perpendicular axis theorem

PARALLEL AXIS THEOREM

According to the parallel axis theorem, the moment of inertia, of a body about any axis is equal to the moment of inertia, which is a parallel axis through the centre of gravity of the body plus

, where M is the mass of the body and d is the distance between the two axes.

PERPENDICULAR AXIS THEOREM

According to perpendicular axes theorem if the moment of inertia is I about z-axis and the two axes, say x-axis and y-axis are mutually perpendicular to the original axis, then we can state:

REPRESENTATION OF MOMENT OF INERTIA:

We can represent all moment of inertia as:

Here, K is known as radius of gyration about the considered axis.

VALUES OF MOMENT OF INERTIA FOR COMMON BODIES:

TYPES OF INERTIA

Inertia is of three types:
o Inertia of rest,
o Inertia of motion and
o Inertia of direction.

INERTIA OF REST

• The tendency of a body to remain in its state of rest is called inertia of rest.

• For example: When we shake a tree, the fruits or dry leaves fall down from the tree

INERTIA OF MOTION

• The tendency of a body to remain in its state of uniform motion in a straight line is called inertia of motion.
• For example:
o An athlete runs for a certain distance before taking a jump so that his inertia of motion may help him to take a longer jump.
o If a horse which is running fast stops suddenly, the rider is thrown forward if he is not firmly seated

INERTIA OF DIRECTION

• The inability of a body to change its direction of motion by itself is called inertia of direction.
• For example: The force exerted on the steering wheel of a car changes its direction of motion.