## Rajasthan Board RBSE Class 9 Science Notes Chapter 9 Force and Motion

**Scalar and vector quantities**

The physical quantities which have only magnitude and need no idea of direction are called scalar quantities. For example: Speed, mass, time, length, area, volume, density, work, energy etc. are scalar quantities.

Vector quantities:

There are certain quantities which require magnitude as well as direction to express them clearly. That is, the physical quantities which have magnitude as well as direction are called Vector quantities.

Unit Vector:

Unit vector of any \(\vec{A}\) is a vector whose magnitude is unit and direction is the direction of vector \(\vec{A}\) Unit vector’s denoted as \(\hat{\mathrm{A}}\) and is read as A cap

\(\hat{A}=|\overline{A}| \hat{A} \Rightarrow \frac{\overline{A}}{|\overline{A}|}\)

**Motion:**

- An object is said to be in motion, if its position from the reference point, changes with time. We require two measurements, to describe the position of an object.
- The distance of the object from a fixed reference point called origin.
- The angle which the line joining the origin and object makes with a reference axis.

Relative nature of motion:

We take an example to explain the nature of motion . A person with his companions is sitting inside a compartment of a running train. The person is at rest relative to other passengers but the person is in motion with respect to the trees and telegraph poles.

This example shows that the same object may be at rest with respect to one reference point and in motion with respect to another at the same time. So, we can say that motion is actually a relative term.

**Distance and displacement:**

- The distance traveled by an object is actual length of the path, traced by it. Distance is a scalar quantity.
- The minimum straight line distance between the first position and the last position of an object, in a particular direction
- is called displacement. Displacement is a vector quantity.

**Speed:**

Distance traveled by a body in unit time is called speed of body. Unit of speed is meter per second. Speed is a scalar quantity because it gives idea of magnitude only and not of direction.

Speed = \(\frac{\text { Distance travelledby the body }}{\text { Time interval }}, \text { i.e., } \mathrm{S}=\frac{d}{t}(\mathrm{m} / \mathrm{sec})\)

**Velocity:**

• The displacement of a moving body per unit time in a definite direction is called its velocity. Its unit is also meter per second. It is a vector quantity.

Velocity \(=\frac{\text { Displacement Vector }}{\text { Time Interval }} \text { or } \overline{v}=\frac{\overline{d}}{t}\)

Difference between Speed and Velocity:

Speed | Velocity |

1. The distance covered by a moving body per unit time is called speed.
2. Speed is scalar quantity. It has magnitude only, direction is not required. |
1. The distance covered by a moving body per unit time, in a definite direction is called velocity.
2. Velocity is a vector quantity. It has a meaning both with magnitude and direction. |

Uniform acceleration:

When the change in the velocity of a moving body is the same per unit interval of time, then its acceleration is said to be uniform. Such motion is said to be a uniformly accelerated motion.

Non-Uniform acceleration:

When the change in velocity of a moving body is not the same per unit interval of time, then its acceleration is said to be non uniform acceleration and the moving body is said to be moving with non-uniform acceleration.

**Graphical representation of motion:**

Velocity time graph:

When a body moves with a constant velocity in a straight line:

Let us consider a body moving with a constant velocity, v. Since velocity remain constant with time, therefore, velocity-time graph is obtained as a straight line parallel to x axis. Let distance covered by the body moving with constant velocity v in time t is s. then the s = v × t.

In the adjoining-figure, let OC represents time t, on x-axis and OA represents velocity v, on y-axis. Therefore, Area of rectangle (OABC) = OC× OA = v × t. This area called the area under the velocity-time graph, represents the total distance traveled by the body in time t.

When a body is moving with uniform acceleration. How to derive the three equations of motion from velocity time graph?:

The velocity time graph for a body under uniform acceleration is shown in figure.

Let initial velocity of the body = u

Final velocity of the body = v

Time taken by the body = t

Acceleration of the body = a

Derivation of the first equation of motion:

According to definition:

Acceleration of a body = Rate of change of velocity, i.e., slope of velocity time graph.

The velocity time graph for a uniformly accelerated body is given by the straight line AB. So, accelerated of the body is equal to the slope of the line AB.

Acceleration = Slope of line

\(A B=\frac{B D}{A D}=\frac{B C-D C}{A D}\)

From the velocity time graph

BC = v, DC = OA = u, AD = OC = t

\(a=\frac{v-u}{t} \Rightarrow a t=v-u \Rightarrow v=u+a t\) This is the first equation of motion

Derivation of the second equation of motion:

Under uniform acceleration, from figure, one can write

Distance travelled (s) = Area of trapezium OABC

s = Area of the triangle ABD + Area of rectangle OADC

\(\frac{1}{2} \times \mathrm{AD} \times \mathrm{BD}+\mathrm{OC} \times \mathrm{OA}\)

Here, AD = OC = t and BD = BC – DC = v – u

BD = u + at – u = at and OA = u

\(s=\frac{1}{2} \times t \times a t+t \times u \Rightarrow s=u t+\frac{1}{2} a t^{2}\) This is the second equation of motion

Derivation of the third equation of motion:

Distance travelled, s = Area of the trapezium OABC

\(s=\frac{1}{2}(\text { sum of the parallel sides }) \times\) perpendicular distance between the two parallel sides

\(s=\frac{1}{2} \times(\mathrm{OA}+\mathrm{BC}) \times \mathrm{OC} \quad s=\frac{1}{2}(u+v) \times t\)

But, \(a=\frac{v-u}{t} \Rightarrow t=\frac{v-u}{a} \Rightarrow s=\frac{1}{2}(v+u) \times \frac{v-u}{a}=\frac{v^{2}-u^{2}}{2 a}\) = \(s=\frac{v^{2}-u^{2}}{2 a} \Rightarrow v^{2}-u^{2}=2 \mathrm{as}\) This is the third equation of motion.

When a body is moving with non-uniform acceleration:

In case of non-uniform motion, the velocity-time graph is shown in figures (i) and (ii).

In figure (i), in time f; velocity increases from zero to v. Here, graph AB shows the body is moving with the positive acceleration, whereas BC shows the body in negative acceleration.

In figure (ii) the body is moving with positive acceleration in first 2 seconds and in between 2 to 6 seconds the body is moving with constant velocity, i.e., acceleration is zero and from 6 to 8 seconds the body is moving with negative acceleration, i.e., retardation. After 8 seconds, the velocity will become zero.

Force and its units:

- In simple words, we can say push or pull is called force.
- On applying force, a stationary body sets in motion and the velocity of the moving body or its direction of motion or both can change.Hence, force is an agent which sets a stationary body in motion or can change the velocity of a moving body. Force is a vector quantity.

Units of Force:

In M.K.S system, the unit of force is Newton. In C.G.S system, the unit of force is dyne. In F.P.S system, the unit of force is poundal.

Balanced and Unbalanced Forces:

In the game of tug of war when the two teams pull with equal effort, the position of the rope is not changed at all. The forces exerted by two teams in this case are balanced. Only when one of the team pulls harder, it is able to pull the weaker team towards it. In this case, forces are unbalanced. Unbalanced forces acting on an object can changes its speed or direction of motion.

**Newton’s laws of motion:**

Scientist Newton propounded three laws of motion:

- First law of motion: According to this law, a body remains at rest or in the state of continuous uniform motion, until and unless it is compelled by an external force to change its state of rest or uniform motion.
- Second Law of motion: The acceleration produced in the body by the action of force acting on it is directly proportional to the force and inversely proportional to the mass of the body. In other words, Newton’s second law of motion can be stated as: The rate of change of momentum is equal to the force applied to the body and the change in momentum always takes place in the direction of force.
- Third Law of Motion: According to this law, “For every action there is an equal and opposite reaction” (Law of action and reaction).

Inertia: Inertia is the property of a body due to which it resists a change in its state of rest or of uniform motion.

**Momentum:
**The vector product of mass and velocity of a moving body is called momentum.

Momentum = Mass of body x Velocity

\(\vec{p}=m \vec{v}\)

Law of conservation of momentum:

- The law states that, in absence of external forces the momentum of the system is conserved. It means that total momentum before the collision is equal to the total momentum after the collision.
- Suppose two bodies A and B of masses m
_{1}and m_{2 }are moving in a straight line in the same direction with velocities u_{1 }and u_{2}, velocity of the body A is more than that of B. So, both the bodies will collide, after sometime. Let velocity of the bodies A and B after collision become v_{1 }and v_{2}, respectively. Let the time of collision equal to t.

If during the collision, the force exerted by body A on body B is F_{1} and force exerted by body B on body A is F_{2}. From second law of motion.

For the body A, \(F_{1} t=m_{1}\left(v_{1}-u_{1}\right)\) …. (i)

For the body B, \(\mathrm{F}_{2} t=m_{2}\left(v_{2}-u_{2}\right)\) …..(ii)

From the third law of motion, F_{1}and F_{2}are equal, but their direction will be opposite. Hence

\(F_{1}=-F_{2}\) …….(iii)

From equation (i), (ii) and (iii), we have \(m_{1}\left(v_{1}-u_{1}\right)=-m_{2}\left(v_{2}-u_{2}\right)\)

Or \(m_{1} v_{1}-m_{1} u_{1}=-m_{2} v_{2}+m_{2} u_{2} \text { or } m_{1} u_{1}+m_{2} u_{2}=m_{1} v_{1}+m_{2} v_{2}\)

i.e. Total momentum before collision = Total momentum after the collision This is what we call the law of conservation of momentum

**Friction:**

- When a body slides or rolls over any surface, a force will develop between the surfaces in contact, which tends to oppose the motion. This force is known as force of friction. The effect of frictional force is more on rough, uneven surfaces than on smooth, even surfaces.
- Bodies moving in air or in liquids also feel the force of friction (though less than that on solid surfaces).
- Initially, when we try to push a body on a surface, it does not move, so long as the force is small, because the frictional force developed is more than the applied force. On gradually increasing the force, a stage will come, when the body begins to move. The maximum force opposing the motion of a body on a surface, when it is just on the verge of moving (by overcoming the frictional force) is called limiting friction. When a body moves sliding on the surface, the force of friction between them is called Sliding Friction. When the body moves on the surface rolling on it (like a ball or wheels) the friction between their surfaces in contact is called rolling friction. Rolling friction is much less than sliding friction.
- Frictional force is a necessary evil. It has many advantages as well as disadvantages. We use methods to increase or decrease the frictional force according to necessity.

**Thrust and Pressure:**

- Thrust: Thrust is the total force acting perpendicular to the surface of a body.
- S.l. unit of thrust: Since thrust is a force. Its S.l. unit is same as that of force, i.e., Newton (N). It is a vector quantity.
- Pressure: Pressure is defined as the force acting perpendicular per unit area of the surface or in other words, we can say that, thrust per unit area is called pressure.

Pressure \(\propto \frac{1}{\text { area }}\) ……(i)

Pressure (P) ∝ Force (F) ……..(ii)

From (i) and (ii), we have

\(P=K \cdot \frac{F}{A}\) ,where K is proportionality constant

When K = 1, F = 1N, A = 1m^{2}, and P = 1 Pascal, then Force (Thrust) = Pressure × Area

Unit of pressure = \(=\frac{N}{m^{2}}, \text { i.e., } N m^{-2}\)

1 N m^{-2}is also known as 1 Pascal (Pa)

1 N m^{-2}= 1 Pascal

i.e., Unit of pressure is also pascal (Pa).

**Buoyant Force:**

- The upward force extended by a liquid on the body which is immersed in the liquid is known as buoyant force.
- The tendency of a fluid to exert an upward force on an object placed in it is called buoyancy.
- Factors on which buoyand force depends:
- The size or volume of the body immersed in a fluid.
- The density of the fluid in which the body is immersed.

**Archimedes’ Principle:**

When a body is wholly or partially immersed in a liquid at rest, it experiences an upthrust (or an apparent loss in weight), is equal to the weight of the liquid displaced by the body. This is the statement of Archimedes principle.

**Applications of Archimedes principle:**

- Designing of ships and submarines.
- Designing lactometers, to test the purity of milk.
- Designing hydrometer, to find the densities of liquid.

**Density:**

Mass per unit volume of a body is known as density.

density \(=\frac{\text { Mass }}{\text { Volume }}\)

\(D=\frac{M}{V}=\frac{k g}{m^{3}}\)

∴ Unit of density is kg.m^{-3}

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