LAWS OF MOTION
Frequently Asked Questions
Newton's First Law states that a body remains at rest or in uniform straight-line motion unless acted upon by a net external force; this is called the law of inertia.
Newton's Second Law states that the rate of change of momentum of a body is proportional to the applied net force and takes place in the direction of the force, \(\vec{F} = \frac{d\vec{p}}{dt} = m\vec{a}\).
Newton's Third Law states that for every action there is an equal and opposite reaction; forces always occur in pairs acting on different bodies.
Inertia is the property of a body by virtue of which it resists any change in its state of rest or uniform motion in a straight line.
Linear momentum \(\vec{p}\) of a body is defined as the product of its mass and velocity, \(\vec{p} = m\vec{v}\).
Impulse of a force is the product of force and the time for which it acts and is equal to the change in momentum, \(I = F\Delta t = \Delta p\).
In an isolated system with no external force, the total linear momentum of the system remains constant during any interaction.
For constant mass, Newton's Second Law reduces to \(\vec{F}_{net} = m\vec{a}\).
An inertial frame is a reference frame in which Newton's First Law holds, i.e., a frame that is either at rest or moving with uniform velocity.
A non-inertial frame is a reference frame that is accelerating with respect to an inertial frame, in which fictitious or pseudo forces must be introduced to apply Newton's laws.
Pseudo force is an apparent force introduced in a non-inertial frame of reference, given by \(\vec{F}_{pseudo} = -m\vec{a}_{frame}\), acting opposite to the acceleration of the frame.
Normal reaction is the contact force exerted by a surface on a body, acting perpendicular to the surface.
Friction is the contact force that opposes the relative motion or the tendency of relative motion between two surfaces in contact.
Static friction acts between surfaces at rest relative to each other and can vary up to a limiting value, while kinetic friction acts when surfaces slide over each other with relative motion.
Limiting friction is the maximum value of static friction just before the body begins to move relative to the surface.
Kinetic friction is given by \(f_k = \mu_k N\), where \(\mu_k\) is the coefficient of kinetic friction and \(N\) is the normal reaction.
Static friction satisfies \(f_s \leq \mu_s N\), where \(\mu_s\) is the coefficient of static friction.
The coefficient of friction is the ratio of limiting friction to normal reaction, \(\mu = \frac{f_{lim}}{N}\); it is dimensionless.
Angle of repose is the maximum angle of an inclined plane with the horizontal at which a body just begins to slide; \(\tan\theta = \mu_s\).
Static friction adjusts its magnitude to match the applied force (up to its limit) so that the body remains at rest, hence it is called self-adjusting.
Rolling friction is the resistive force that opposes the motion of a rolling body on a surface and is generally much smaller than sliding friction.
Sliding friction is generally greater than rolling friction for the same pair of surfaces.
Centripetal force is the net force required to keep a body in uniform circular motion and is directed towards the center, \(F_c = \frac{mv^2}{r}\).
A car turning on a level road and a cyclist taking a turn on a flat track use friction between tyres and road to provide centripetal force.
Banking of roads means raising the outer edge of a curved road to provide a component of normal reaction towards the center to act as centripetal force.
For a frictionless banked curve, \(\tan\theta = \frac{v^2}{rg}\), where \(\theta\) is banking angle, \(v\) speed, \(r\) radius, and \(g\) acceleration due to gravity.
Apparent weight is the normal reaction experienced by a body, which may differ from true weight \(mg\) in accelerating frames like lifts.
In a lift accelerating up: \(N = m(g + a)\); accelerating down: \(N = m(g - a)\); in free fall: \(N = 0\).
Due to inertia, the lower body moves with the bus while the upper body tends to remain at rest, so the passenger feels a backward push relative to the bus.
Due to inertia of motion, the upper body continues to move forward while the bus and feet come to rest, causing the passenger to lurch forward.
Mass is the quantity of matter and measure of inertia (scalar, SI unit kg), while weight is the gravitational force acting on mass, \(W = mg\) (vector, SI unit newton).
Contact forces act via physical contact (normal, friction, tension), while non-contact forces act at a distance (gravitational, electrostatic, magnetic).
Tension is the pulling force transmitted along a stretched string, rope, or cable, equal in magnitude at all points for a light, inextensible string in equilibrium.
For two masses \(m_1\) and \(m_2\) pulled by force \(F\), acceleration \(a = \frac{F}{m_1 + m_2}\) and tension can be found from \(T = m_1 a\) or \(T = F - m_2 a\).
Atwood's machine consists of two masses connected over a light frictionless pulley; acceleration \(a = \frac{(m_2 - m_1)g}{m_1 + m_2}\) (assuming \(m_2 > m_1\)).
Pulling at an angle reduces normal reaction and hence friction, while pushing at an angle increases normal reaction and friction, so pulling is easier.
Isolate the body, represent it as a point or simple shape, show all external forces with correct directions and labels, and choose a convenient coordinate system.
For each body, write \(\sum F_x = m a_x\) and \(\sum F_y = m a_y\) from its FBD, then solve the system of equations for unknowns like acceleration and tension.
Newton's Second Law quantitatively relates force, mass, and acceleration, while the First and Third Laws can be derived or understood using it, so it is considered the fundamental law.
Before firing, total momentum is zero; after firing, forward momentum of bullet equals backward momentum of gun so total remains zero, causing gun recoil.
Rockets expel gases backward at high speed; by conservation of momentum, the rocket gains forward momentum even in vacuum.
In a sudden stop, car decelerates but passenger tends to continue forward due to inertia; seatbelts provide restraining force, reducing injuries.
Friction between foot and ground prevents slipping, providing reaction force that pushes the body forward as the person exerts backward force on the ground.
A particle is in equilibrium when the vector sum of all forces on it is zero, so its acceleration is zero.
For a particle: \(\sum F_x = 0\) and \(\sum F_y = 0\); for rigid bodies (beyond this chapter), \(\sum \tau = 0\) is also used.
If \(F > \mu_k N\), acceleration \(a = \frac{F - \mu_k N}{m}\); if \(F \leq \mu_s N\), the block does not move and \(a = 0\).
For a block of mass \(m\) on an incline angle \(\theta\), components of weight are \(mg\sin\theta\) down the plane and \(mg\cos\theta\) normal; friction opposes motion along the plane.
On a smooth (frictionless) incline of angle \(\theta\), acceleration down the plane is \(a = g\sin\theta\).
Key topics include Newton's laws, friction (graphs and coefficients), circular motion and banking, connected bodies and pulleys, pseudo force, and conservation of momentum with numericals.
Draw FBDs for each mass, define directions of acceleration, apply \(F = ma\) equations consistently, use string constraints for equal accelerations where required, then solve algebraically.
Typical questions include calculating friction on horizontal and inclined planes, proving angle of repose relation, explaining self-adjusting nature, and comparing static and kinetic friction.
Laws of motion form the base for advanced mechanics topics like work-energy, circular motion, gravitation, and are heavily used in higher-level problem solving in competitive exams.
Area under a force–time graph gives impulse, which equals change in momentum; numericals often ask to compute impulse from such graphs.
Clearly indicate directions with sign convention, use unit vectors for components, and avoid mixing scalar and vector forms in the same equation.
