10th Science Unit 1 Laws of Motion Book Back Answers

Inertia is the property of a body to resist change in its state of rest or uniform motion.

This resistance is measured by the quantity of matter in the body, i.e. its mass. Weight and $g$ change from place to place, but inertia does not.

Impulse $J = F \times t$.

By Newton's second law $F = \Delta p / t$, so $\Delta p = F \times t = J$.

Hence impulse equals the change in momentum (unit N s or kg m s$^{-1}$).

Newton's third law says for every action there is an equal and opposite reaction, acting on two different bodies.

This holds for a book resting on a table (rest) as well as for a swimmer pushing water or a fired gun recoiling (motion).

Slope of a momentum–time graph $=$ change in momentum / time $= \Delta p / t$.

By Newton's second law, the rate of change of momentum is the force.

The turning (rotating) effect of a force is torque, τ = F × d, which makes a body rotate about an axis.

In cycling, the rider presses the pedal at a distance from the axle; this force produces a moment (torque) that turns the crank and the wheels.

Swimming (third law), tennis and hockey (impulse/force of a hit) do not primarily use the turning effect of force.

From $F = mg$, $g = F/m$, so the unit of $g$ is newton per kilogram.

$1\ \mathrm{N\,kg^{-1}} = 1\ \mathrm{kg\,m\,s^{-2}/kg} = 1\ \mathrm{m\,s^{-2}}$.

$1\ \mathrm{kgf} = 1\ \mathrm{kg} \times 9.8\ \mathrm{m\,s^{-2}} = 9.8\ \mathrm{N}$.

Since $1\ \mathrm{N} = 10^5$ dyne, $9.8\ \mathrm{N} = 9.8 \times 10^5 = 98 \times 10^4$ dyne.

Mass is the quantity of matter contained in a body — a basic property of the body.

It does not depend on $g$ or on the planet, so on the new planet the mass is still M kg (only the weight changes).

$g = GM/R^2$. If $R$ becomes $R/2$ with $M$ unchanged, $g' = GM/(R/2)^2 = 4g$.

Weight $W' = m(4g) = 4W$, an increase of $3W$, i.e. 300% increase.

Hot gases ejected backward from the nozzle give a huge momentum; an equal and opposite reaction pushes the rocket forward — Newton's third law.

As no net external force acts, the momentum of the rocket–gas system stays constant — law of conservation of linear momentum.

Force is an external push or pull that produces or tries to produce motion in a static body.

Without force, a body at rest stays at rest, so no displacement occurs.

The lower part of the body stops with the vehicle, but the upper part tends to continue moving forward.

This resistance to a change in the state of motion is inertia of motion.

By convention, rotation in the clockwise direction is taken as negative.

Rotation in the anticlockwise direction is taken as positive.

A gear is a circular wheel with teeth around its rim.

It changes the speed of rotation of a wheel by changing the torque and transmits power.

Weight $W = mg$.

$W = 100 \times 9.8 = 980$ N.

Momentum is conserved only when no net external force acts on the system.

If an external unbalanced force acts, the total momentum of the system changes.

So the word 'always' makes the statement wrong.

Apparent weight is the reaction force R of the surface on the person.

In a lift moving up with acceleration a, $R = m(g+a)$ (more than actual weight); moving down, $R = m(g-a)$ (less than actual weight); in free fall $R = 0$.

Only when the lift is at rest or moves with uniform velocity (a = 0) is $R = mg$, the actual weight.

Weight $W = mg$, and $g \propto 1/R^2$.

The geometric radius of the Earth is maximum at the equator and minimum at the poles, so g is maximum at the poles.

Hence a body weighs more at the poles than at the equator.

The turning effect (moment) of a force is $\tau = F \times d$, where d is the distance from the axis of rotation.

A long handle gives a larger d, so the same force produces a larger moment.

Therefore turning a nut is easier with a long-handled spanner.

Gravity of the Earth acts on the space station and the astronauts; that is what keeps them in orbit.

Because of their huge orbital velocity, the astronauts and the station are falling freely around the Earth with the same acceleration.

In free fall the reaction force R = m(g – g) = 0, so they feel weightless.

a. Newton's I law – Stable equilibrium of a body: a body stays at rest (or in uniform motion) unless an external force acts on it.

b. Newton's II law – Law of force: F = ma helps us measure the amount of force, so it is called the law of force.

c. Newton's III law – Flying nature of bird: the bird pushes the air down (action) and the air pushes the bird up (reaction).

d. Law of conservation of linear momentum – Propulsion of a rocket: hot gas is ejected backward with a large momentum and the rocket moves forward with equal momentum.

Assertion: By the principle of moments, when a rigid body is in equilibrium the algebraic sum of clockwise moments equals the algebraic sum of anticlockwise moments ($F_1 \times d_1 = F_2 \times d_2$). So the assertion is true.

Reason: The principle of conservation of linear momentum states that there is no change in the linear momentum of a system as long as no net external force acts on it. So the reason is also true.

But the two statements deal with different ideas — one is about turning effect (torque) in equilibrium, the other is about momentum of a system. Hence the reason is not the explanation of the assertion.

Assertion: As we move to a higher altitude from the Earth's surface, the value of g reduces, and when we go deep below the surface also g reduces (g is zero at the centre of the Earth). So the assertion is true.

Reason: From $g = \dfrac{GM}{R^2}$, the acceleration due to gravity depends only on the mass $M$ and radius $R$ of the Earth — it does not depend on the mass of the object placed on it. So the reason is false.

Hence the assertion is true but the reason is false.

Inertia is the inherent property of a body to resist any change in its state of rest or state of uniform motion, unless acted upon by an external unbalanced force.

Inertia of rest: the resistance of a body to change its state of rest. Example: leaves fall when a tree branch is shaken.

Inertia of motion: the resistance of a body to change its state of motion. Example: an athlete runs some distance before jumping.

Inertia of direction: the resistance of a body to change its direction of motion. Example: we lean sideways when a car takes a sharp turn.

Based on the direction in which the forces act, forces are classified into two types.

Like parallel forces: two or more forces of equal or unequal magnitude acting along the same direction, parallel to each other.

Unlike parallel forces: two or more equal or unequal forces acting along opposite directions, parallel to each other. Example: tug of war.

Given: $F_1 = 15$ N and $F_2 = 5$ N acting in opposite directions (unlike parallel forces).

For parallel unequal forces acting in opposite directions, $F_{net} = F_1 - F_2$.

$F_{net} = 15 - 5 = 10$ N.

The resultant is directed along the direction of the greater force, i.e. along the 15 N force.

Mass is the quantity of matter contained in a body; weight is the gravitational force exerted on the body due to gravity, $W = mg$.

Mass is a scalar quantity; weight is a vector quantity directed towards the centre of the Earth.

SI unit of mass is kilogram (kg); SI unit of weight is newton (N).

Mass of a body remains the same everywhere; weight varies from place to place because it depends on 'g' (more at the poles, less at the equator, less on the Moon).

Two equal and unlike parallel forces applied simultaneously at two distinct points, whose lines of action do not coincide, constitute a couple.

A couple produces no translatory motion (resultant is zero) but causes rotation of the body. This rotating effect is called moment of a couple.

Moment of a couple = one of the forces × perpendicular distance between the lines of action of the two forces, $M = F \times S$.

Its unit is newton metre (N m) in SI and dyne cm in CGS. Examples: turning a tap, winding a screw.

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.

In other words, at equilibrium the algebraic sum of the moments of all the individual forces about any point is zero.

Mathematically, $F_1 \times d_1 = F_2 \times d_2$.

Newton's second law states that 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.

$F \propto \dfrac{mv - mu}{t}$, and taking the proportionality constant as 1, $F = \dfrac{m(v-u)}{t}$.

Since $a = \dfrac{v-u}{t}$, we get $F = ma$, i.e. force = mass × acceleration.

This law helps to measure the amount of force, so it is also called the 'law of force'.

Tightening a screw needs a large turning effect, i.e. a large moment of force.

Moment of force $\tau = F \times d$, where d is the perpendicular distance from the axis of rotation.

A long handle gives a larger d, so for the same applied force the torque produced is much greater.

Hence the screw can be tightened easily with less effort using a long-handled spanner.

A moving ball has momentum; to catch it the fielder must reduce this momentum to zero, i.e. produce a change in momentum (impulse).

Impulse $J = F \times t = \Delta p$, so for a fixed change in momentum, force and time are inversely related.

By drawing his hands backwards, the fielder increases the time of contact t.

Therefore the force experienced by his hands is smaller and his hands are not hurt.

It is wrong to think that there is no gravity in the orbiting space station.

The astronaut and the space station are actually falling freely around the Earth due to their huge orbital velocity.

Since both have the same acceleration (equal to g), they are in the free fall condition.

In free fall the reaction force of the floor on the astronaut is zero, R = m(g – g) = 0, i.e. apparent weight is zero.

This state of weightlessness makes the astronaut appear to float inside the space shuttle.

Given: masses in the ratio $m_1 : m_2 = 3 : 4$, acceleration of the bigger mass $a_2 = 12\,\mathrm{ms^{-2}}$, same force $F$ on both.

Formula (Newton's second law): $F = ma$. Since $F$ is the same, $m_1 a_1 = m_2 a_2$.

So $a_1 = \dfrac{m_2}{m_1} \times a_2 = \dfrac{4}{3} \times 12$.

Result: $a_1 = 16\,\mathrm{ms^{-2}}$ — the lighter body gets the larger acceleration.

Given: mass $m = 1\,\mathrm{kg}$, speed before hitting the floor $u = 10\,\mathrm{ms^{-1}}$ (downward), and after a perfectly elastic rebound $v = 10\,\mathrm{ms^{-1}}$ upward.

Formula: change in linear momentum $\Delta p = mv - mu$.

Taking the downward direction as negative: $\Delta p = (1 \times 10) - (1 \times (-10))$.

$\Delta p = 10 + 10 = 20\,\mathrm{kg\,ms^{-1}}$, directed upward (opposite to the initial motion).

Given: $F_1 = 140\,\mathrm{N}$, $d_1 = 40\,\mathrm{cm} = 0.4\,\mathrm{m}$, $F_2 = 40\,\mathrm{N}$, $d_2 = ?$

Formula: to unscrew the same nut the moment of the force must be the same, $F_1 \times d_1 = F_2 \times d_2$.

Substituting: $140 \times 0.4 = 40 \times d_2$, so $56 = 40\,d_2$.

$d_2 = \dfrac{56}{40} = 1.4\,\mathrm{m}$.

Given: $M_1 : M_2 = 2 : 3$ and $R_1 : R_2 = 4 : 7$.

Formula: acceleration due to gravity $g = \dfrac{GM}{R^2}$.

Ratio: $\dfrac{g_1}{g_2} = \dfrac{M_1}{M_2} \times \left(\dfrac{R_2}{R_1}\right)^2 = \dfrac{2}{3} \times \left(\dfrac{7}{4}\right)^2$.

$\dfrac{g_1}{g_2} = \dfrac{2}{3} \times \dfrac{49}{16} = \dfrac{98}{48} = \dfrac{49}{24}$.

Result: $g_1 : g_2 = 49 : 24$.

1. Measuring dimensions of heavenly bodies: Dimensions of the heavenly bodies can be measured using the gravitation law. The mass of the Earth, radius of the Earth, acceleration due to gravity, etc. can be calculated with higher accuracy.

2. Discovering new bodies: It helps in discovering new stars and planets.

3. Mass of a star from 'Wobble': One of the irregularities in the motion of stars, called 'Wobble', leads to disturbance in the motion of a nearby planet. Under this condition the mass of the star can be calculated using the law of gravitation.

4. Geotropism: It helps to explain the germination of roots, which is due to geotropism — the property of a root responding to gravity.

5. Predicting paths: It helps to predict the path of astronomical bodies.

Given: masses $m_1 = 8$ kg and $m_2 = 2$ kg in contact on a smooth (frictionless) surface; applied force $F = 15$ N.

Both blocks move together, so total mass $M = 8 + 2 = 10$ kg.

By Newton's second law, acceleration of the system $a = \dfrac{F}{M} = \dfrac{15}{10} = 1.5$ m s$^{-2}$.

The only force acting on the 2 kg block is the contact (push) force from the 8 kg block. So $F_2 = m_2 \times a$.

$F_2 = 2 \times 1.5 = 3$ N.

Given: kinetic energy of truck = kinetic energy of bike; $m_{truck} = 4\,m_{bike}$.

Kinetic energy $KE = \dfrac{1}{2}mv^2$ and momentum $p = mv$, so $KE = \dfrac{p^2}{2m}$, giving $p = \sqrt{2m\,KE}$.

Since KE is the same for both, $p \propto \sqrt{m}$.

$\dfrac{p_{truck}}{p_{bike}} = \sqrt{\dfrac{m_{truck}}{m_{bike}}} = \sqrt{\dfrac{4m}{m}} = \sqrt{4} = 2$.

Hence $p_{truck} : p_{bike} = 2 : 1$ (the heavier truck carries the greater momentum).

By Newton's first law (law of inertia), a passenger moving along with a vehicle possesses inertia of motion. When a sudden brake is applied, the vehicle stops but the body of the passenger continues to move forward and is thrown ahead. A fastened seat belt holds the body back and prevents this forward throw; a helmet protects the head that is thrown forward.

By Newton's second law, force is the rate of change of momentum, $F = \Delta p / t$. In a sudden stop or crash the momentum changes in a very short time, so a very large impulsive force acts on the body.

The seat belt and the padding inside the helmet stretch/compress slightly and increase the time of impact $t$. Since the change in momentum (impulse $J = \Delta p$) is fixed, a longer time means a much smaller force on the body — the same reason a cricket fielder pulls his hands back while catching a ball.

By Newton's third law, whatever force the body applies on the vehicle or on the road during impact, an equal and opposite reaction acts on the body. The helmet spreads this reaction force over a larger area of the skull and absorbs part of it, so the head escapes serious injury.