Grade 09 Physics: How Forces Affect Motion

Fundamentals of Balanced and Unbalanced Forces

• A force is an external push or pull acting upon an object that can change its state of rest, change its uniform motion in a straight line, alter its direction, or deform its shape and size.

• Balanced Forces:

  • When the resultant (net) sum of all external forces acting simultaneously on a body is zero ($F_{\text{net}} = 0$), the forces are defined as balanced forces.

  • Balanced forces cannot change the state of rest of an object.

  • Balanced forces cannot set a stationary body into motion.

  • Balanced forces cannot change the speed or direction of a moving object (they produce zero acceleration, $a = 0\,m/s^2$).

  • Balanced forces can alter the shape and dimensions (size) of an object (e.g., compressing a rubber ball from opposite sides with equal force).

• Unbalanced Forces:

  • When the resultant of all forces acting simultaneously on a body is non-zero ($F_{\text{net}} \neq 0$), the forces are defined as unbalanced forces.

• An unbalanced force accelerates a body ($a \neq 0\,m/s^2$), changing its speed, its direction of motion, or both.

  • Continued application of an unbalanced force results in continuous change in velocity or continuous change in direction.

• Vector Force System Resolution:   

  

  • Vertical force balance: An upward force of $25\,N$ directly opposes a downward force of $25\,N$, yielding a net vertical force of $F_{y,\text{net}} = 25\,N - 25\,N = 0\,N$.

  • Horizontal force balance: A horizontal force of $15\,N$ pushes to the right, and an additional horizontal force of $5\,N$ acts toward the right. The net horizontal force is $F_{x,\text{net}} = 15\,N + 5\,N = 20\,N$ directed to the right.

  • Resultant status: Because $F_{\text{net}} = 20\,N \neq 0$, the system experiences unbalanced forces.

Newton's First Law of Motion and the Principle of Inertia

• Newton's First Law of Motion:

  • A body remains in its state of rest or of uniform motion in a straight line unless compelled to change that state by an applied unbalanced external force.

  • This principle establishes that uniform velocity does not require a sustained driving force when friction and resistive forces are absent.

  • If an object of mass $2\,kg$ slides with a constant velocity of $4\,m/s$ on a frictionless horizontal table, the force required to keep it moving at that identical velocity is strictly $0\,N$.

  • Similarly, for an object of mass $6\,kg$ sliding on a frictionless horizontal plane at a uniform velocity of $8\,m/s$, maintaining its motion requires $0\,N$ of external force.

• Inertia:

  • Inertia is the inherent resistance of an object to any change in its velocity or state of motion.

  • Mass is the quantitative measure of inertia: an object with larger mass possesses greater inertia and offers greater resistance to changes in its motion.

• Inertia of Rest:

  • Tendency of a body to remain stationary until acted upon by an external unbalanced force.

  • Coin-Card-Tumbler Experiment:     

    

  • A five-rupee coin rests on a stiff card placed over an open glass tumbler.

  • When the card is flicked rapidly with a finger, a sharp horizontal force acts on the card alone, propelling it away.

  • The coin remains at rest due to its inertia of rest; once the supporting card is removed, gravity pulls the coin vertically downward into the tumbler.

  • If replaced with a heavier coin, the greater mass provides greater inertia of rest, causing it to resist displacement even more effectively and drop cleanly into the tumbler.

  • Luggage on a Bus Roof:

  • When a stationary bus abruptly accelerates forward, unrestrained luggage tends to remain stationary due to inertia of rest and slides backward off the roof.

• Inertia of Motion:

  • Tendency of a moving object to continue traveling at uniform speed along a straight trajectory.

  • Liquid Surge in Tankers:

  • A water tanker filled to $\frac{2}{3}$ of its height moves at uniform speed.

  • When brakes are applied suddenly, the base of the vehicle decelerates, but the water continues traveling forward due to inertia of motion, surging toward the front wall.

  • Decelerating Vehicles:

  • Luggage tied to a bus roof resists deceleration when brakes are applied, sliding forward toward the front of the bus unless secured with ropes.

• Inertia of Direction:

  • Tendency of an object to maintain its straight-line heading.

  • When a motorcar negotiates a sharp turn at high speed, passengers are thrown outward toward the outside of the curve because their bodies attempt to continue moving along the original linear tangent.

Linear Momentum and Newton's Second Law of Motion

• Linear Momentum ($p$):

  • Momentum measures the total quantity of motion contained within a body.

  • Defined as the product of mass ($m$) and velocity ($v$):     $p = m \times v$

  • SI Unit: $\text{kg}\cdot\text{m/s}$ (or $\text{kg}\,\text{m}\,\text{s}^{-1}$).

  • Momentum is a vector quantity having the same direction as velocity.

  • At any given instant, momentum depends directly on mass, velocity, and inertia, but is mathematically independent of instantaneous acceleration.

  • Example Calculation (Walking Man):

  • For a man of mass $m = 70\,kg$ walking with uniform velocity $v = 2\,m/s$:       $p = 70\,kg \times 2\,m/s = 140\,\text{kg}\cdot\text{m/s}$

  • Example Calculation (Decelerating Mass):

  • For a body of mass $m = 5\,kg$ with speed decreasing from $u = 20\,m/s$ to $v = 0.20\,m/s$:       $p_{\text{initial}} = 5\,kg \times 20\,m/s = 100\,\text{kg}\cdot\text{m/s}$       $p_{\text{final}} = 5\,kg \times 0.20\,m/s = 1.0\,\text{kg}\cdot\text{m/s}$       $\Delta p = p_{\text{final}} - p_{\text{initial}} = 1.0 - 100 = -99\,\text{kg}\cdot\text{m/s}$

  • The momentum decreases by $99\,\text{kg}\cdot\text{m/s}$.

• Newton's Second Law of Motion:

  • The rate of change of momentum of a body is directly proportional to the applied unbalanced force and takes place in the direction in which the force acts:     $F \propto \frac{\Delta p}{\Delta t}$

  • Mathematical Formulation:     $p_1 = m \times u$     $p_2 = m \times v$     $\Delta p = m \times (v - u)$     $F = k \times \frac{m \times (v - u)}{\Delta t} = k \times m \times a$

  • Setting the unit constant of proportionality $k = 1$ yields:       $F = m \times a$

  • Expressing acceleration:       $a = \frac{F}{m}$

  • Acceleration is directly proportional to the applied force ($a \propto F$) and inversely proportional to the mass of the object ($a \propto \frac{1}{m}$).

• Comparative Force Requirements:

  • Case 1 ($3\,kg$ at $5\,m/s^2$ vs. $4\,kg$ at $4\,m/s^2$):     $F_1 = 3\,kg \times 5\,m/s^2 = 15\,N$     $F_2 = 4\,kg \times 4\,m/s^2 = 16\,N$

  • A greater force is required for accelerating the $4\,kg$ mass at $4\,m/s^2$.

  • Case 2 ($2\,kg$ at $5\,m/s^2$ vs. $4\,kg$ at $2\,m/s^2$):     $F_A = 2\,kg \times 5\,m/s^2 = 10\,N$     $F_B = 4\,kg \times 2\,m/s^2 = 8\,N$

  • Accelerating the $2\,kg$ mass requires more force ($10\,N > 8\,N$).

• Impulse and Force Reduction Mechanisms:

  • Rearranging the second law demonstrates that force varies inversely with contact duration for a fixed momentum change:     $F = \frac{\Delta p}{\Delta t}$

  • Goalkeeper Pulling Hands Backward:

  • When catching a shot on goal, pulling hands backward increases the time interval $\Delta t$ over which the ball's momentum is brought to zero.

  • Increasing $\Delta t$ decreases the rate of change of momentum, thereby decreasing the impact force exerted by the ball onto the goalkeeper's hands.

  • High Jumper Sand Pit or Cushion Landing:

  • Athletes land on soft sand or cushions upon completing a high jump to prolong the time taken to come to a complete stop ($\Delta t$).

  • This minimizes the rate of change of momentum and reduces the peak impact force on bones and joints, preventing injury.

• Experimental Verification: Acceleration-Force Relationship:   

  

  • The plot of acceleration ($y$-axis) versus applied force ($x$-axis) yields a straight line passing through the origin $(0,0)$.

  • The constant linear slope represents the inverse mass of the system:     $\text{Slope} = \frac{a}{F} = \frac{1}{m}$

  • Using graph coordinates $(F = 2\,N, a = 1.0\,m/s^2)$ or $(F = 4\,N, a = 2.0\,m/s^2)$:     $m = \frac{F}{a} = \frac{2\,N}{1.0\,m/s^2} = 2\,kg$

  • If applied force is doubled on the same body ($m = 10\,kg$, from $20\,N$ to $40\,N$), acceleration doubles proportionally from $2\,m/s^2$ to $4\,m/s^2$.

• Kinematic Data and Net Force Deduction:

  • Consider an object moving according to the following distance-time data:

• Analysis of Motion:

  • Over equal successive intervals of $\Delta t = 1\,s$, the displacements are $2\,m$, $14\,m$, $38\,m$, $74\,m$, $122\,m$, $182\,m$, and $254\,m$.

  • Because distance increases cubically with time ($s(t) = 2 \times t^3$), the velocity ($v(t) = 6 \times t^2$) and acceleration ($a(t) = 12 \times t$) are continuously increasing.

  • The motion is non-uniformly accelerated motion.

• Deduction of Net Force:

  • An increasing rate of change of distance with time confirms increasing velocity and positive acceleration.

  • By $F_{\text{net}} = m \times a$, a non-zero, continuously increasing net unbalanced force must be acting on the object.

Newton's Third Law of Motion and Interaction Pairs

• Statement of Newton's Third Law:

  • To every action, there is always an equal and opposite reaction.

  • When object $A$ exerts a force on object $B$, object $B$ simultaneously exerts an equal magnitude force on object $A$ directed in the opposite sense:     $F_{AB} = -F_{BA}$

• Essential Properties of Action-Reaction Pairs:

  • Action and reaction forces act on two entirely different bodies simultaneously.

  • Because they act on different objects, action-reaction forces never cancel each other out.

  • Equilibrium cancellation only occurs when two equal and opposite forces act on the same physical body.

  • Newton's third law applies universally to all force classifications (gravitational, electrostatic, magnetic, contact, tension) and holds true whether interacting bodies are static or dynamic.

• Physical Scenarios and Case Studies:

  • Recoil of Firearms and Cannons:

  • When a bullet is fired from a gun, expanding gases push the bullet forward (action).

  • The bullet exerts an equal and opposite backward reaction force on the gun, causing it to recoil.

  • The bullet accelerates at a vastly higher rate than the gun because its mass is significantly smaller ($a = \frac{F}{m}$).

  • Rotary Lawn Water Sprinkler:     

    

  • Water is forced out through curved nozzles at high velocity (action force on water).

  • The discharging streams exert an equal and opposite reactive thrust against the nozzle tips (reaction force on nozzles).

  • This backward reactive force creates a net torque around the central axis, rotating the sprinkler arms automatically.

  • Rope-Tension Interaction (Kent Budgett Demonstration):     

    

  • Top picture: Kent Budgett pulls a rope attached to a wall, producing an in-line scale reading of $600\,N$.

  • Bottom picture: Kent pulls a rope attached to an elephant, producing an in-line scale reading of $600\,N$.

  • A spring balance measures internal tension transmitted through the line, which requires equal forces pulling outward on both ends.

  • In both instances, Kent pulls with a force of $600\,N$; the immovable wall reacts with $600\,N$, and the resisting elephant reacts with $600\,N$.

  • Walking on a Floating Boat:

  • A person standing on a boat in still water steps forward toward the shore (action: feet push the boat backward).

  • The boat exerts an equal reaction force pushing the person toward shore, while the boat itself drifts backward away from the shore.

  • Two Skaters Tossing a Ball:

  • Skater 1 pushes a ball forward (action on ball); the ball pushes skater 1 backward, causing them to roll backward on frictionless skates.

  • Skater 2 catches the ball, absorbing its forward momentum; the ball exerts a forward force on skater 2, causing skater 2 to roll backward.

  • Both skaters roll away from each other, increasing their separation distance.

  • Swimmer Propulsion:

  • A swimmer pushes water backward with their hands and feet (action force on water).

  • The displaced water pushes the swimmer forward with an equal and opposite force (reaction force on swimmer).

  • Collision Disparity (Firefly vs. Windshield):

  • When a flying firefly strikes the windshield of a bus, both experience an identical magnitude of impact force ($F_{\text{firefly on bus}} = F_{\text{bus on firefly}}$).

  • The firefly suffers extreme damage because its tiny mass experiences an immense acceleration ($a = \frac{F}{m}$), whereas the bus's enormous mass renders its deceleration undetectable.

Rocket Propulsion and Orbital Dynamics

• Rocket Launch Mechanism:   

  

  • During initial ascent, rocket engines burn fuel and expel mass exhaust gas downward at high velocity (action force).

  • By Newton's Third Law of Motion, escaping gases exert an equal and opposite reaction force pushing the rocket upward (thrust).

  • Because upward thrust exceeds the downward force of gravity and atmospheric drag, the net force acting on the rocket is non-zero ($F_{\text{net}} > 0$), producing upward acceleration by Newton's Second Law ($a = \frac{F_{\text{net}}}{m}$).

• Orbital Mechanics and Satellite Motion:

  • Once positioned in a stable circular orbit outside Earth's atmosphere, no forward engine thrust is required.

  • Tangential Speed Retention: Newton's First Law of Motion governs the satellite's forward motion; in the frictionless vacuum of space, it retains its tangential speed without propulsion.

  • Centripetal Force: Earth's gravity acts as the centripetal force directed toward the planetary center, constantly changing the satellite's velocity vector to maintain a curved orbit rather than traveling in a straight line:     $F_c = \frac{m \times v^2}{r}$

  • Weightlessness: An astronaut inside an orbiting satellite experiences weightlessness due to being in a continuous state of free fall toward Earth at the exact same rate as the spacecraft.

Multi-Body Mechanics and Connected Systems

• Single-System Method for Coupled Bodies:   

  

  • Two bodies of masses $m_1$ and $m_2$ connected by a light, inextensible cable on a frictionless surface can be analyzed as a single unified system of mass $M = m_1 + m_2$.

  • Internal forces, such as string tension ($T$), occur in equal and opposite pairs ($T$ pulling forward on $m_2$, $T$ pulling backward on $m_1$) and cancel out when considering the combined system boundary.

  • System acceleration depends exclusively on the net external force:     $a = \frac{F_{\text{ext}}}{m_1 + m_2}$

  • Two-Block Problem Calculation:

  • Given $m_1 = 3\,kg$, $m_2 = 2\,kg$, and external horizontal pull $F = 15\,N$ on Box 1:       $a = \frac{15\,N}{3\,kg + 2\,kg} = \frac{15}{5} = 3.0\,m/s^2$

• Warehouse Conveyor Analysis:   

  

  • Context: In an automated facility, a mechanical arm applies an external horizontal force of $F = 200\,N$ to Platform 1 ($m_1 = 60\,kg$), which connects to Platform 2 ($m_2 = 40\,kg$) via a steel cable on a frictionless magnetic track.

  • System Acceleration:     $M_{\text{total}} = m_1 + m_2 = 60\,kg + 40\,kg = 100\,kg$     $a = \frac{F}{M_{\text{total}}} = \frac{200\,N}{100\,kg} = 2\,m/s^2$

  • Evaluation of Differing Acceleration Claim:

  • The concern that internal cable tension causes the front and rear platforms to accelerate at different rates is physically incorrect.

  • Because the platforms are joined by an inextensible cable, they form a rigid kinematic unit and must accelerate at the exact same rate ($a = 2\,m/s^2$).

  • Internal versus External Force on Individual Body:

  • When isolating Platform 2 ($m_2 = 40\,kg$) as an independent free-body diagram, the primary forward-driving external force acting upon it is the tension ($T$) of the connecting cable:       $T = m_2 \times a = 40\,kg \times 2\,m/s^2 = 80\,N$

  • For Platform 1 ($m_1 = 60\,kg$), the net force driving it is:       $F_{\text{net}, 1} = F - T = 200\,N - 80\,N = 120\,N$       $a_1 = \frac{120\,N}{60\,kg} = 2\,m/s^2$

Systematic Solutions to Question Bank Problems

• Multiple Choice Questions:

  • Question 1: Vertical forces cancel ($25\,N - 25\,N = 0\,N$); horizontal forces add ($15\,N + 5\,N = 20\,N$). Result: $F_{\text{net}} \neq 0$: Unbalanced forces. Correct option: (d).

  • Question 2: Moving at constant velocity on a frictionless surface requires zero acceleration ($a = 0\,m/s^2$), hence $F = 0\,N$. Correct option: (b).

  • Question 3: Pulling hands back increases impact time, which decreases the rate of change of momentum and reduces the force exerted on the hands. Correct option: (b) [also fundamentally described by (d)].

  • Question 4: Weightlessness inside an orbiting satellite results from the state of continuous free fall. Correct option: (d).

  • Question 5: Forces that change an object's state of rest or uniform linear motion are unbalanced forces. Correct option: (c).

  • Question 6: Statements (a), (b), and (c) correctly characterize balanced forces; hence none of them is incorrect. Correct option: (d).

  • Question 7: Inertia causes an object to resist any change in its state of motion. Correct option: (c).

  • Question 8: Statement (i) is false (an unbalanced force is required to accelerate); statements (ii) and (iii) are true. Correct option: (b).

  • Question 9: Mass $6\,kg$, uniform velocity $8\,m/s$, no friction requires $F = 0\,N$ to maintain motion. Correct option: (a).

  • Question 10: Inertia is a quantitative measure of mass. Correct option: (b).

  • Question 11: Applying brakes causes water in the tanker to move forward due to inertia of motion. Correct option: (b).

  • Question 12: $F_1 = 3 \times 5 = 15\,N$; $F_2 = 4 \times 4 = 16\,N$. Force is greater for the $4\,kg$ mass at $4\,m/s^2$. Correct option: (d).

  • Question 13: Goalkeeper retracting hands reduces the force exerted by the ball on hands. Correct option: (b).

  • Question 14: Recoil of a cannon is explained by Newton's third law of motion. Correct option: (c).

  • Question 15: Momentum ($p = m \times v$) is independent of instantaneous acceleration. Correct option: (d).

  • Question 16: The scale measures line tension ($600\,N$), which requires equal opposing reactions from wall and elephant; Kent pulls with $600\,N$ in both cases. Correct option: (c).

  • Question 17: Momentum is the quantitative physical measure of motion. Correct option: (d).

  • Question 18: Change in momentum from $20\,m/s$ to $0.20\,m/s$ for a $5\,kg$ body: $\Delta p = (5 \times 0.20) - (5 \times 20) = 1 - 100 = -99\,\text{kg}\cdot\text{m/s}$. Correct option: (b).

  • Question 19: $F_1 = 2 \times 5 = 10\,N$; $F_2 = 4 \times 2 = 8\,N$. Accelerating the $2\,kg$ mass requires more force. Correct option: (a).

  • Question 20: Walking toward the shore pushes the boat backward into the water; the boat moves away from the shore. Correct option: (a).

  • Question 21: In a single connected system, internal forces do not need to be considered when calculating total acceleration. Correct option: (b).

  • Question 22: $a = \frac{F}{m_1 + m_2} = \frac{15}{3 + 2} = 3.0\,m/s^2$. Correct option: (b).

• Assertion and Reason Questions:

  • Question 1: Assertion is true; Reason is true; but Reason is not the proper explanation of why balanced forces preserve state (which is due to net vector cancellation). Correct option: (b).

  • Question 2: Assertion is true; Reason is true; friction opposing motion explains why a bicycle slows down when pedalling stops. Correct option: (a).

  • Question 3: Assertion is false (an object maintains uniform motion without any unbalanced force; an unbalanced force causes continuous acceleration); Reason is true. Correct option: (d).

  • Question 4: Assertion is true; Reason is true; inertia of direction directly explains why passengers are thrown outward on sharp turns. Correct option: (a).

  • Question 5: Assertion is true; Reason is true; the relationship $F \propto \frac{\Delta p}{\Delta t}$ provides the quantitative definition and measurement of force ($F = ma$). Correct option: (a).

  • Question 6: Assertion is true; Reason is true; the rate of change of momentum is directly proportional to applied force, dictating force magnitude. Correct option: (a).

  • Question 7: Assertion is true ($F_{\text{net}} = 0 \implies a = 0$); Reason is false (acceleration directly depends on force, $a = \frac{F}{m}$). Correct option: (c).

  • Question 8: Assertion is true (action-reaction forces are identical); Reason is false (the bus has far greater mass than the firefly). Correct option: (c).

  • Question 9: Assertion is true (Newton's third law applies across all forces); Reason is false (applies equally to bodies at rest). Correct option: (c).

  • Question 10: Assertion is true; Reason is true; equal and opposite action-reaction dictates identical forces on bullet and gun. Correct option: (a).

• Numerical and Applied Problems:

  • Applied Problem 1 (Accelerating Car):

  • Given: $m = 1200\,kg$, $u = 0\,m/s$, $v = 20\,m/s$, $t = 10\,s$.

  • Acceleration:       $a = \frac{v - u}{t} = \frac{20 - 0}{10} = 2\,m/s^2$

  • Force applied:       $F = m \times a = 1200\,kg \times 2\,m/s^2 = 2400\,N$

  • Applied Problem 2 (Skateboarder Mass):

  • Given: $u = 1\,m/s$, $v = 4\,m/s$, $t = 2.5\,s$, $F = 72\,N$.

  • Acceleration:       $a = \frac{v - u}{t} = \frac{4 - 1}{2.5} = \frac{3}{2.5} = 1.2\,m/s^2$

  • Combined mass:       $m = \frac{F}{a} = \frac{72\,N}{1.2\,m/s^2} = 60\,kg$

  • Numerical Question 6 (Acceleration from Force):

  • Given: $m = 2\,kg$, $F = 1\,N$.

  • Acceleration:       $a = \frac{F}{m} = \frac{1\,N}{2\,kg} = 0.5\,m/s^2$