10th Standard Science: Unit 1 - Laws of Motion Study Notes

Newton’s First Law and the Concept of Inertia

• Newton’s First Law of Motion: This law states that every body continues to be in its state of rest or the state of uniform motion along a straight line unless it is acted upon by some external force. This law provides two fundamental definitions: it defines 'force' and it defines 'inertia.'

• Inertia: The inherent property of a body to resist any change in its state of rest or the state of uniform motion unless it is influenced upon by an external unbalanced force is known as inertia.

• Types of Inertia:

  • Inertia of Rest: The resistance of a body to change its state of rest is called inertia of rest.

  • Inertia of Motion: The resistance of a body to change its state of motion is called inertia of motion.

  • Inertia of Direction: The resistance of a body to change its direction of motion is called inertia of direction.

• Examples of Inertia:

  • Inertia of Motion: An athlete runs some distance before jumping because this will help him jump longer and higher.

  • Inertia of Direction: When you make a sharp turn while driving a car, you tend to lean sideways.

  • Inertia of Rest: When you vigorously shake the branches of a tree, some of the leaves and fruits are detached and they fall down.

Definition and Classification of Force

• Force: Force is an external effort in the form of push or pull, which:

  1. Produces or tries to produce the motion of a static body.

  2. Stops or tries to stop a moving body.

  3. Changes or tries to change the direction of motion of a moving body.

• Vector Nature of Force: Force has both magnitude and direction. Therefore, it is classified as a vector quantity.

• Types of Forces Based on Direction:

  • Like Parallel Forces: Two or more forces of equal or unequal magnitude acting along the same direction, parallel to each other, are called like parallel forces.

  • Unlike Parallel Forces: If two or more equal forces or unequal forces act along opposite directions parallel to each other, then they are called unlike parallel forces.

Resultant Force and Equilibrium

• Resultant Force: When several forces act simultaneously on the same body, the combined effect of the multiple forces can be represented by a single force, which is termed the ‘resultant force.’ It is equal to the vector sum (adding the magnitude of the forces with their direction) of all the forces.

• Balanced Forces: If the resultant force of all the forces acting on a body is equal to zero ($F_{net} = 0$), then the body will be in equilibrium. Such forces are called balanced forces.

• Unbalanced Forces: If the resultant force is not equal to zero, then it causes the motion of the body due to unbalanced forces.

  • Examples: Drawing water from a well, force applied with a crowbar, forces on a weight balance, etc.

• Equilibrant: A system can be brought to equilibrium by applying another force, which is equal to the resultant force in magnitude but opposite in direction. Such a force is referred to as the ‘Equilibrant.’

• Table 1.1: Action of Forces

  • Parallel forces acting in the same direction: $F_{net} = F_1 + F_2$

  • Parallel unequal forces acting in opposite directions: $F_{net} = F_1 - F_2$ (if F_1 > F_2) or $F_{net} = F_2 - F_1$ (if F_2 > F_1). The $F_{net}$ is directed along the greater force.

  • Parallel equal forces acting in opposite directions in the same line of action ($F_1 = F_2$): $F_{net} = F_1 - F_2 = 0$.

Linear Momentum

• Definition: The impact of a force is greater if the velocity and the mass of the body are higher. To quantify this impact exactly, linear momentum is defined. It measures the impact of a force on a body.

• Mathematical Expression: The product of mass and velocity of a moving body gives the magnitude of linear momentum. It acts in the direction of the velocity of the object.

$p = m \times v$

• Properties:

  • Linear momentum is a vector quantity.

  • It helps to measure the magnitude of a force.

  • Units:

    • SI system: $kg\,m\,s^{-1}$

    • C.G.S system: $g\,cm\,s^{-1}$

Rotating Effect of Force and Torque

• Observations on Rotation: A door can be easily opened or closed when force is applied far from the hinges (fixed edge). The turning effect of the applied force is more when the distance between the fixed edge and the point of application of force is greater.

• Axis of Rotation: The axis of the fixed edge about which the door is rotated is called the ‘axis of rotation.’

• Point of Rotation: If one end of a rod is fixed to a point and a tangential force is applied at the other end, the rod turns about the fixed point, called the ‘point of rotation.’

• Moment of the Force (Torque): The rotating or turning effect of a force about a fixed point or fixed axis is called the moment of the force about that point or torque ($\tau$).

• Formula for Torque: It is measured by the product of the force ($F$) and the perpendicular distance ($d$) between the fixed point (or fixed axis) and the line of action of the force.

$\tau = F \times d$

• Characteristics of Torque:

  • Torque is a vector quantity.

  • It acts along the direction perpendicular to the plane containing the line of action of force and the distance.

  • SI unit: $N\,m$.

Couple and Moment of a Couple

• Definition of Couple: Two equal and unlike parallel forces applied simultaneously at two distinct points constitute a couple. The lines of action of the two forces do not coincide.

• Motion Produced: A couple does not produce any translatory motion because the resultant force is zero. However, it results in the rotation of the body.

• Moment of a Couple: The rotating effect of a couple is the product of any one of the forces and the perpendicular distance between the lines of action of the two forces.

$M = F \times S$

• Examples: Turning a tap, winding or unwinding a screw, spinning of a top.

• Units of Moment of a Couple:

  • SI system: $N\,m$

  • CGS system: $dyne\,cm$

• Sign Conventions:

  • Positive: If the body is rotated in the anti-clockwise direction.

  • Negative: If the body is rotated in the clockwise direction.

Applications of Torque and the Principle of Moments

• Applications of Torque:

  1. Gears: Circular wheels with teeth that change rotation speed by changing torque and help transmit power.

  2. Seesaw: A heavier person can be lifted by a lighter person if the heavier person moves closer to the pivot point (fulcrum), decreasing the torque they exert.

  3. Steering Wheel: A small steering wheel allows for easy maneuvering by transferring torque to the wheels with less effort.

• Principle of Moments: When a number of like or unlike parallel forces act on a rigid body and the body is in equilibrium, the algebraic sum of the moments in the clockwise direction is equal to the algebraic sum of the moments in the anticlockwise direction. Alternatively, the algebraic sum of the moments of all individual forces about any point is equal to zero.

$Moment\ in\ clockwise\ direction = Moment\ in\ anticlockwise\ direction$

$F_1 \times d_1 = F_2 \times d_2$

Newton’s Second Law of Motion

• Definition: According to this law, “the force acting on a body is directly proportional to the rate of change of linear momentum of the body and the change in momentum takes place in the direction of the force.”

• Derivation of the Law of Force:

  • Let $m$ be the mass of a body with initial speed $u$.

  • After time $t$, velocity changes to $v$ due to force $F$.

  • Initial momentum ($P_i$) = $mu$

  • Final momentum ($P_f$) = $mv$

  • Change in momentum ($\Delta p$) = $mv - mu$

  • Force is proportional to the rate of change of momentum:     $F \propto \frac{mv - mu}{t}$     $F = \frac{k \times m(v - u)}{t}$

  • Since $k = 1$ and acceleration $a = \frac{v - u}{t}$:

$F = m \times a$

• Force Requirements: No external force is required to maintain uniform velocity. Force is required only to produce acceleration. In uniform circular motion, speed is constant but direction changes, creating centripetal acceleration and requiring centripetal force.

• Units of Force:

  • SI unit: newton (N)

  • CGS unit: dyne

  • Definition of 1 Newton: The amount of force required for a body of mass $1\,kg$ to produce an acceleration of $1\,m\,s^{-2}$. ($1\,N = 1\,kg\,m\,s^{-2}$).

  • Definition of 1 Dyne: The amount of force required for a body of mass $1\,g$ to produce an acceleration of $1\,cm\,s^{-2}$. ($1\,dyne = 1\,g\,cm\,s^{-2}$).

  • Conversion: $1\,N = 10^5\,dyne$.

• Gravitational Units of Force:

  • SI: kilogram force ($kg\,f$). $1\,kgf = 9.8\,N$.

  • CGS: gram force ($g\,f$). $1\,gf = 980\,dyne$.

Impulse

• Definition: A large force acting for a very short interval of time is called an 'Impulsive force.' The product of force ($F$) and time ($t$) is known as 'impulse' ($J$).

$J = F \times t$

• Relationship to Momentum: According to Newton's Second Law ($F = \frac{\Delta p}{t}$), hence $J = \Delta p$. Impulse is equal to the magnitude of the change in momentum.

• Units of Impulse: $kg\,m\,s^{-1}$ or $N\,s$.

• Achieving Change in Momentum:

  1. A large force acting for a short period of time.

  2. A smaller force acting for a longer period of time.

• Examples:

  • Automobiles: Fitted with springs and shock absorbers to reduce jerks on uneven roads by increasing the time of impact.

  • Cricket: A fielder pulls his hands back while catching a ball to experience a smaller force over a longer interval, resulting in lesser impulse/pain.

Newton’s Third Law of Motion

• Definition: For every action, there is an equal and opposite reaction. They always act on two different bodies.

• Mathematical Representation: If body A applies force $F_A$ on body B, body B reacts with force $F_B$ on body A.

$F_B = -F_A$

• Examples:

  • Birds Flying: Wings push air down (Action); air pushes bird up (Reaction).

  • Swimming: Person pushes water back (Action); water pushes person forward (Reaction).

  • Firing a Bullet: Gun pushes bullet forward (Action); bullet pushes gun backward (Recoil/Reaction).

Principle of Conservation of Linear Momentum

• Definition: There is no change in the linear momentum of a system of bodies as long as no net external force acts on them.

• Proof:

  • Two bodies with masses $m_1, m_2$ and initial velocities $u_1, u_2$ collide for time $t$.

  • After collision, velocities are $v_1, v_2$.

  • Force on body B due to A: $F_A = \frac{m_2(v_2 - u_2)}{t}$

  • Force on body A due to B: $F_B = \frac{m_1(v_1 - u_1)}{t}$

  • By Newton's III law ($F_B = -F_A$):     $\frac{m_1(v_1 - u_1)}{t} = -\frac{m_2(v_2 - u_2)}{t}$     $m_1v_1 - m_1u_1 = -m_2v_2 + m_2u_2$

$m_1v_1 + m_2v_2 = m_1u_1 + m_2u_2$

• This confirms that the algebraic sum of momentum after collision equals the sum before collision in the absence of external forces.

Rocket Propulsion

• Mechanism: Based on the Law of Conservation of Linear Momentum and Newton’s III Law of Motion.

• Process: Propellant tanks are filled with fuel (liquid or solid). When fired, fuel burns and hot gas is ejected at high speed from the nozzle (huge momentum). This produces an equal and opposite reaction force in the combustion chamber, projecting the rocket forward.

• Mass Variation: As the rocket moves, mass decreases as fuel burns. Since there is no net external force, linear momentum is conserved. Decrease in mass leads to a gradual increase in velocity.

• Escape Velocity: The velocity at which the rocket can escape the Earth's gravitational pull.

Gravitation and Acceleration Due to Gravity

• Newton’s Universal Law of Gravitation: Every particle of matter in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

$F = \frac{G \times m_1 \times m_2}{r^2}$

• Universal Gravitational Constant ($G$): In SI units, its value is $6.674 \times 10^{-11}\,N\,m^2\,kg^{-2}$.

• Acceleration Due to Gravity ($g$): The acceleration produced in a body due to the Earth's gravitational force.

  • Mean value on Earth's surface: $9.8\,m\,s^{-2}$.

  • Velocity changes by $9.8\,m\,s^{-1}$ every second during free fall.

• Relation between $g$ and $G$:

  • By Newton's law: $F = \frac{GMm}{R^2}$

  • By Newton's second law ($F = mg$):     $mg = \frac{GMm}{R^2}$

$g = \frac{GM}{R^2}$

• Mass of the Earth ($M$): Calculated by rearranging the formula to $M = \frac{gR^2}{G}$.

  • Approximate value: $M = 5.972 \times 10^{24}\,kg$.

Variation and Distribution of 'g'

• Geographic Variation: Since $g \propto \frac{1}{R^2}$, its value changes with the Earth's radius. Radius is maximum at the equator and minimum at the poles.

  • Value of g: Maximum at poles; minimum at the equator.

• Altitude and Depth: The value of $g$ reduces as you move to higher altitudes or deeper below the Earth’s surface.

• Earth's Center: The value of $g$ is zero at the center of the Earth.

Mass and Weight

• Mass: The basic property of a body defined as the quantity of matter contained within it. Unit: kilogram ($kg$). It remains constant regardless of location.

• Weight: The gravitational force exerted on a body. $Weight = m \times g$. Unit: newton ($N$).

  • Weight is a vector quantity directed toward the center of the Earth.

  • Weight varies with location because $g$ varies.

• Weight on the Moon: The acceleration due to gravity on the moon is $1.625\,m\,s^{-2}$, which is roughly $0.1654$ times that of Earth.

  • Example: A 60 kg person weighs $588\,N$ on Earth ($60 \times 9.8$) but only $97.5\,N$ on the Moon ($60 \times 1.625$).

Apparent Weight and Weightlessness

• Apparent Weight ($R$): The weight of a body acquired due to the action of gravity and other external forces. In a lift, the actual weight is $W = mg$ acting down, and the reaction force ($R$) of the floor is the apparent weight acting up.

• Table 1.2: Apparent Weight in a Moving Lift:

  • Moving Upward (acceleration $a$): $R = m(g + a)$; R > W (Feels heavier).

  • Moving Downward (acceleration $a$): $R = m(g - a)$; R < W (Feels lighter).

  • At Rest ($a = 0$): $R = mg$; $R = W$.

  • Free Fall ($a = g$): $R = m(g - g) = 0$; $R = 0$ (Weightlessness).

• Weightlessness of Astronauts: Astronauts in orbiting space stations are not floating because of lack of gravity, but because they are in a state of free fall. Both the station and the astronauts have the same acceleration ($a = g$), resulting in an apparent weight of zero ($R = 0$).

Applications of Newton’s Law of Gravitation

1. Measuring Heavenly Bodies: Accurate calculation of mass, radius, and gravity of planets and stars.

2. Discovery: Helps in discovering new stars and planets.

3. Wobble: Used to calculate the mass of stars by analyzing irregularities in their motion caused by nearby planets.

4. Geotropism: Explains the germination of roots responding to gravity.

5. Astro-Prediction: Helps predict the paths of astronomical bodies.

Solved Problems

• Problem 1 (Momentum): Calculate velocity for $m = 5\,kg$ and $p = 2.5\,kg\,m\,s^{-1}$.

  • $V = \frac{p}{m} = \frac{2.5}{5} = 0.5\,m\,s^{-1}$.

• Problem 2 (Torque): Force of $40\,N$ applied at $90\,cm$ ($0.9\,m$) from hinges.

  • $M = F \times d = 40 \times 0.9 = 36\,N\,m$.

• Problem 3 (Gravity variation): Height where $g' = \frac{g}{4}$.

  • $\frac{g}{g'} = (\frac{R + h}{R})^2 \rightarrow \frac{g}{g/4} = 4 = (1 + \frac{h}{R})^2$

  • $2 = 1 + \frac{h}{R} \rightarrow h = R$. The object is at twice the radius from the center ($R' = 2R$).