Force and Laws of Motion - Practice Flashcards

8.1 Balanced and Unbalanced Forces

• Forces are pushes or pulls that can change the state of motion or shape of an object.
• When two forces acting on an object are equal in magnitude and opposite in direction, they are balanced forces and do not change the object's state of rest or uniform motion along a straight line.
• If the forces are not balanced (i.e., their magnitudes are different or directions are not opposite), an unbalanced force acts and changes the motion or direction of the object.
• Friction is a force that acts between surfaces in contact and opposes relative motion. It can balance a pushing force on a rough surface, preventing motion until the pushing force exceeds friction.
• Examples from daily life include a box on a rough floor: at low push, friction balances the push and the box does not move; increase push, overcome friction, and the box starts to move.
• When moving, friction continues to oppose motion. To keep motion, we must apply a continuing unbalanced force.
• In practical terms, a uniform velocity requires the net external force to be zero (balanced forces). A nonzero net external force produces acceleration.

8.2 First Law of Motion

• Galileo’s inclined plane experiments showed that, in the absence of external forces like friction, an object would move with constant speed on a frictionless path. A marble rolls down an inclined plane gaining speed due to gravity; up the rise, speed decreases.
• If the incline is symmetric and frictionless, the marble would rise back to the original height and, with zero unbalanced force, would continue indefinitely at that height (in practice friction prevents this).
• From Galileo, Newton formulated the First Law of Motion: An object remains in a state of rest or in uniform motion in a straight line unless compelled to change that state by an applied force.
• This tendency to resist changes in motion is inertia. The first law is also called the Law of Inertia.
• Everyday experiences (e.g., car braking, passengers lurching forward; sudden starts/stops) illustrate inertia.
• Inertia is more pronounced in more massive objects; heavier objects have greater inertia.

8.3 Inertia and Mass

• Inertia is the resistance of any physical object to a change in its state of motion or rest.
• Mass is a quantitative measure of inertia. Heavier objects resist changes in motion more than lighter ones.
• Observations show that it is easier to push an empty box than a box full of books; a lighter ball is easier to accelerate than a heavier one with the same push.
• The following questions illustrate inertia conceptually:
  • Which has more inertia: a rubber ball vs. a stone of the same size? a bicycle vs. a train? a five-rupees vs. one-rupee coin?
  • How many times does velocity change in a sequence of kicks and passes, and who supplies the force at each step?
  • Why do leaves detach when a branch is vigorously shaken?
  • Why do you lurch forward when a moving bus brakes and lurch backward when it accelerates from rest?
• Inertia is quantitatively captured by mass, with the common relation that more massive objects have greater inertia.

8.4 Second Law of Motion

• The observed fact: accelerating a body requires a greater force for greater accelerations, and momentum plays a key role in this relationship.
• Momentum p of an object is defined as the product of its mass and velocity: p = mv. Momentum has both magnitude and direction (same as velocity). The SI unit is \mathrm{kg\,m\,s^{-1}}.
• The Second Law states that the rate of change of momentum of an object is proportional to the applied unbalanced force, in the direction of the force. In its common form, this leads to F = ma when force is constant and mass is constant.
• The law also implies that a force changes momentum: the impulse imparted to an object is the product of force and the time for which the force acts: \mathbf{J} = \Delta \mathbf{p} = \mathbf{F}\Delta t.
• If a body of mass m is moving with initial velocity u and ends with velocity v after time t under a constant force, then
  • Change in momentum: \Delta p = mv - mu = m(v - u)
  • Rate of change of momentum: \frac{\Delta p}{\Delta t} = \frac{m(v-u)}{t} = a m
  • Therefore, F = ma and in terms of momentum: F\,t = \Delta p = m(v-u).
• The impulse-momentum connection explains everyday phenomena: catching a fast ball is easier if you increase the time over which you stop it (longer impulse lowers peak force); cushioning landings reduces peak force by increasing stopping time.
• The unit of force is newton: 1\ \text{N} = 1\ \mathrm{kg\,m\,s^{-2}}; the newton is defined as the force that produces an acceleration of 1\ \mathrm{m\,s^{-2}} in a body of mass 1\ \mathrm{kg}.
• Practical examples:
  • A fielder catching a fast-moving ball by pulling hands back to increase stopping time reduces the force on the hands.
  • A high jump uses a cushioned bed or sand to increase stopping time and reduce impact force.

8.4.1 Mathematical Formulation of Second Law of Motion

• Consider an object of mass m moving along a straight line with initial velocity u and final velocity v in time interval t under constant force F.
• Change in momentum: \Delta p = mv - mu.
• Since force is proportional to the rate of change of momentum, we have:
  • F \propto \frac{\Delta p}{\Delta t} = \frac{m(v-u)}{t}
  • Introducing proportionality constant to define the SI unit of force, we obtain:
  • F = ma where a = \frac{v-u}{t} is the acceleration.
• The constant of proportionality is chosen to make the unit equal to Newton; hence, F = ma.
• The impulse form: F\,t = \Delta p = m(v-u).
• Examples illustrate calculations:
  • Example 8.1: A constant force acts on a 5 kg mass for 2 s, increasing velocity from 3 to 7 m/s.
  • Compute force: F = m\frac{v-u}{t} = 5\frac{7-3}{2} = 10\ \text{N}.
  • If the same force acts for 5 s, final velocity: v = u + \frac{F}{m}t = 3 + \frac{10}{5}\cdot 5 = 13\ \text{m/s}.
  • Example 8.2: Compare forces required for two cases: m1=2\ \text{kg}, a1=5\ \text{m s}^{-2} and m2=4\ \text{kg}, a2=2\ \text{m s}^{-2}.
  • F1 = m1 a1 = 2\cdot 5 = 10\ \text{N}; F2 = m2 a2 = 4\cdot 2 = 8\ \text{N}. Hence, the lighter mass with higher acceleration requires greater force in this comparison.
  • Example 8.3: A car of mass 1000 kg moving at 30 m/s comes to rest in 4 s.
  • Stop-force: F = m\frac{v-u}{t} = 1000\cdot\frac{0-30}{4} = -7500\ \text{N} (magnitude 7500 N, opposite to motion).
  • Example 8.4: Two masses 0.50 kg and 0.25 kg are pushed with the same 5 N force and then joined.
  • Individual accelerations: a1 = 10 m/s^2, a2 = 20 m/s^2. But when combined (total mass 0.75 kg), acceleration becomes a = \frac{F}{m} = \frac{5}{0.75} \approx 6.67\ \text{m s}^{-2}.
  • Example 8.5: A 20 g ball on a table comes to rest from 20 cm/s in 10 s due to friction.
  • Acceleration: a = \frac{v-u}{t} = \frac{0 - 0.20}{10} = -0.02\ \text{m s}^{-2}.
  • Frictional force: F = ma = (0.02\ \text{kg})(-0.02\ \text{m s}^{-2}) = -4\times 10^{-4}\ \text{N}.

8.5 Third Law of Motion

• For every action, there is an equal and opposite reaction; the action and reaction forces act on two different objects simultaneously and have equal magnitudes but opposite directions.
• Everyday illustrations:
  • Kicking a football: the kick exerts a force on the ball; the ball exerts an equal and opposite force on the foot.
  • Gun recoil: the gun exerts a forward force on the bullet; the bullet exerts an equal and opposite force on the gun; due to greater mass of the gun, its acceleration is smaller.
  • Sailor jumping from a boat: throwing oneself forward propels the boat backward.
• A pair of action–reaction forces can be demonstrated with two connected springs: the force exerted by spring A on B is equal and opposite to the force of B on A.

Quick Takeaways: Key Concepts and Formulas

• First Law (Law of Inertia): An object at rest stays at rest, and an object in uniform straight-line motion stays in that motion unless acted upon by a net external unbalanced force. Inertia is the resistance to change in motion; mass measures inertia.
• Friction opposes motion and can balance pushes, preventing motion until the push exceeds friction.
• Second Law: The rate of change of momentum is proportional to the applied unbalanced force in the direction of the force. In the common form for constant mass: F = ma.
• Momentum: p = mv; momentum is a vector (direction along velocity). Change in momentum equals the impulse: \Delta p = F\Delta t.
• Units: Force in Newtons (N); 1 N = \mathrm{kg\,m\,s^{-2}}.
• Impulse helps explain stopping distance and impact forces: increasing the time over which the force acts reduces peak forces.
• Third Law: Forces come in action–reaction pairs acting on different bodies, equal in magnitude and opposite in direction.

Selected Practice Problems and Examples (from the chapter)

• Example 8.1: A constant force acts on a 5 kg mass for 2 s, changing velocity from 3 m/s to 7 m/s.
  • Force: F = m\frac{v-u}{t} = 5\cdot\frac{7-3}{2} = 10\ \text{N}
  • If force acted for 5 s, final velocity: v = u + \frac{F}{m}t = 3 + \frac{10}{5}\cdot 5 = 13\ \text{m/s}
• Example 8.2: Compare forces for two situations: m1=2\text{ kg}, a1=5\text{ m s}^{-2} and m2=4\text{ kg}, a2=2\text{ m s}^{-2}.
  • F1 = m1a1 = 2\cdot5 = 10\ \text{N}; F2 = m2a2 = 4\cdot2 = 8\ \text{N}; thus F1 > F2.
• Example 8.3: A car of mass 1000 kg moving at 30 m/s stops in 4 s.
  • Brake force: F = m\frac{v-u}{t} = 1000\cdot\frac{0-30}{4} = -7500\ \text{N} (magnitude 7500 N; opposite to motion).
• Example 8.4: A 5 N force acts on two masses (0.50 kg and 0.25 kg) separately with accelerations 10 and 20 m/s^2; if tied together, acceleration is a = \frac{F}{m} = \frac{5}{0.75} \approx 6.67\ \text{m s}^{-2}.
• Example 8.5: Velocity-time graph for a 20 g ball on a table, from 0.20 m/s to 0 in 10 s.
  • Acceleration: a = \frac{0-0.20}{10} = -0.02\ \text{m s}^{-2}
  • Friction force: F = ma = (0.02\ \text{kg})(-0.02\ \text{m s}^{-2}) = -4\times 10^{-4}\ \text{N}

Quick Practice: Conceptual Questions

• If net external unbalanced force is zero, can an object move with nonzero velocity? Yes; it can have constant velocity (Inertia, First Law).
• Why does dust come out when a carpet is beaten? Due to inertia and separation of dust from carpet fibers when the force acts; dust acquires motion with the carpet and is left behind when the carpet stops moving.
• Why tie luggage on the roof of a bus? To prevent it from being displaced by external forces during motion (turbulence, braking, turning).
• Third Law nuance: Action and reaction act on different bodies and may not produce equal accelerations if masses differ.
• Momentum and impulse relations help explain outcomes in collisions and stopping events.

Important Definitions and Units

• Momentum: p = mv, vector with the same direction as velocity. SI unit: \mathrm{kg\,m\,s^{-1}}.
• Force: F = ma; SI unit: Newton (N). 1 N = \mathrm{kg\,m\,s^{-2}}.
• Impulse: \mathbf{J} = \Delta \mathbf{p} = \mathbf{F}\Delta t.
• Acceleration: a = \frac{v-u}{t}.
• Inertia: resistance to change in motion; quantified by mass.

Connections to Foundational Principles and Real-World Relevance

• Newton’s Laws generalize everyday experiences: pushing an object, braking a vehicle, the recoil of firearms, and the behavior of a person jumping from a boat.
• In engineering and safety design, impulse and momentum concepts guide how to design brakes, airbags, seat belts, and protective paddings to optimize stopping times and reduce injury.
• Everyday safety and sports strategies rely on increasing time over which forces act (catching balls, cushions on impact, etc.).