Revision: Laws of Motion Physics HSC Science (General) 11th Standard Maharashtra State Board

"If an object continuously changes its position, it is said to be in motion."

A train or a moving vehicle on a road may travel in a straight line or in the same direction. This motion of an object is called linear motion. An object in linear motion shows displacement along a straight line."

"The motion of an object that does not move in a straight line is called ‘non-linear motion’."

Aristotle's statement: “An external force is required to keep a body in uniform motion”.

"If no force is acting on a body, its velocity does not change, i.e., the body does not accelerate. In other words, if a body is stationary, it will remain stationary. If it is in motion, it will continue moving with the same velocity and in the same direction."

or

"An object continues to remain at rest or in a state of uniform motion along a straight line unless an external unbalanced force acts on it."

or

"Every inanimate object continues to be in its state of rest or of uniform unaccelerated motion unless and until it is acted upon by an external, unbalanced force."

Newton’s second law of motion states that the rate of change of momentum is directly proportional to force applied and takes place in the direction of the force.

"The rate of change of momentum is proportional to the applied force, and the change of momentum occurs in the direction of the force."

"Every action force has an equal and opposite reaction force which acts simultaneously."

The concise law statement is: "To every action (force), there is an equal and opposite reaction (force)."

A non-inertial frame of reference is accelerating. In these frames, you may experience "fictitious" or pseudo-forces.

A frame of reference typically uses a coordinate system (axes and an origin) to track motion. If an object’s coordinates change with time in your chosen system, it is in motion relative to that frame.

An inertial frame of reference obeys Newton’s first law:

A body at rest stays at rest, and a body in motion continues at constant velocity, unless acted on by a net force.

A force is seen to act through direct contact of the objects or via one more object. Such a force is called 'Contact force.'

A force is applied between two objects even if the two objects are not in contact; such a force is called a 'Non-contact force.'

A force that does not follow the conservative force rule, where the work done by or against it depends on the actual path taken.

A force is said to be a conservative force if the work done by or against it is independent of the actual path chosen and depends only on the initial and final positions of the object.

"If there is a decrease in the potential energy (like a body falling down) due to a conservative force, it is entirely converted into kinetic energy. Work done by the force then appears as kinetic energy. Vice versa if an object is moving against a conservative force its kinetic energy decreases by an amount equal to the work done against the force."

“The total momentum of an isolated system is conserved during any interaction.”

For two colliding bodies, the negative of the ratio of the relative velocity of separation to the relative velocity of approach is called the coefficient of restitution.

The source describes a collision as a process where "several objects come together, interact (exert forces on each other) and scatter in different directions."

A collision where the kinetic energy of the entire system is conserved during the collision (along with the linear momentum).

An event where two or more objects come into contact and exert forces on each other, causing changes in their individual momenta while the total momentum of the system remains conserved.

A collision where there is a loss in kinetic energy during the collision, but linear momentum is conserved.

∑KE_initial = ∑KE_final

Where:

• ∑KE_initial = total initial kinetic energy
• ∑KE_final = total final kinetic energy

The exact definition from the source is: "If colliding bodies join together after collision, it is said to be a perfectly inelastic collision."

The negative of the ratio of the relative velocity of separation to the relative velocity of approach.

Mathematically: e = -\[\frac{\text{relative velocity of separation}}{\text{relative velocity of approach}}\]

Perfectly Inelastic Collision: A perfectly inelastic, head-on collision of two bodies of masses m₁ and m₂ with respective initial velocities u₁ and u₂, where they move jointly after the collision, i.e., their final velocity is the same.

A collision in two dimensions (oblique collision) is defined as: a collision where the direction of at least one initial velocity is NOT along the line of impact, requiring analysis using two mutually perpendicular directions—the line of impact and the common tangent at the point of contact.

Oblique or non-head-on collision.

The quantity ‘change in momentum’ is separately named as the Impulse of the force.

The rotational analogue of a force, defined as the quantity that represents the rotational ability of a force. It depends upon the magnitude of the force, the point of application of the force, and the angle between the direction of the force and the line joining the axis of rotation with the point of application.

A couple is a pair of forces that satisfy all of the following:

• They are equal in magnitude (same strength)
• They are opposite in direction
• Their lines of action do not coincide (they act along different straight paths)
  Because of these properties, a couple produces a pure rotation (turning) effect and no net translation (no sliding) of the object.

Mechanical equilibrium occurs when the combined effect of all forces on an object equals zero, resulting in constant velocity or rest.

"Centre of mass is a point about which the summation of moments of masses in the system is zero."

The centre of mass is a hypothetical point at which the entire mass of the body can be assumed to be concentrated.

The centre of gravity is an imaginary location where the body’s whole weight is assumed to be concentrated.

The Centre of Gravity (c.g.) of a body is the point around which the resultant torque due to the force of gravity on the body is zero.

or

The centre of gravity (C.G.) of a body is the point about which the algebraic sum of moments of the weights of all the particles constituting the body is zero. The entire weight of the body can be considered to act at this point, howsoever the body is placed.

\[\vec F\] = m \[\frac{d\vec{\mathrm{v}}}{dt}\] = m\[\vec a\] ... (for constant mass)

Thus, if \[\vec F\] = 0, \[\vec v\] is constant. Hence, if there is no force, velocity will not change. This is nothing but Newton's first law of motion.

General Form: \[\vec F\] =\[\frac{d\vec{p}}{dt}\]

For Constant Mass: \[\vec F\] = m\[\vec a\]

Momentum: \[\vec p\] = m\[\vec v\]

\[\vec{F}=\frac{d\vec{p}}{dt}=\frac{d\left(m\vec{\mathrm{v}}\right)}{dt}\]

Pseudo Force: F_pseudo = −m\[\vec a\]

Where:

• m = mass of the object
• \[\vec a\] = acceleration of the non-inertial reference frame
• The negative sign indicates the force is opposite to the acceleration direction

\[\sum\vec{p}_{initial}\] = \[\sum\vec{p}_{final}\]

Where:

• \[\sum\vec{p}_{initial}\] = total initial momentum of all objects
• \[\sum\vec{p}_{final}\] = total final momentum of all objects

∑KE_initial > ∑KE_final

Where:

• The lost kinetic energy is due to internal friction or vibrational motion of atoms, causing a heating effect
• Linear momentum is still conserved: \[\sum\vec{p}_{initial}\] = \[\sum\vec{p}_{final}\]

Impulse (J) = F × Δt = Δp = m(v - u)

Where:

• J = Impulse
• F = Force applied
• Δt = Time interval for which the force acts
• Δp = Change in momentum
• m = Mass of the object
• v = Final velocity
• u = Initial velocity

Vector Form

\[\vec τ\] = \[\vec r\] × \[\vec F\]

Here:

• \[\vec τ\] = Torque
• \[\vec r\] = Position vector of the point of application of force from the axis of rotation
• \[\vec F\] = applied force

Scalar (Magnitude) Form

τ = r F sin θ

Where:

• r =  Perpendicular distance from the axis of rotation
• F = Magnitude of the force
• θ = The smaller angle between the direcions of \[\vec r\] and \[\vec F\]

• The weight of a body acts through a single point called the centre of gravity (C.G.), where the sum of moments of all particles' weights is zero.
• The position of the C.G. depends on the shape and mass distribution of the body and changes if the body is deformed.
• The C.G. may lie outside the material of the body (e.g., a ring or hollow sphere).
• A body balances when supported exactly at its centre of gravity, as seen in a metre rule or square lamina.
• The C.G. of an irregular lamina can be found by suspending it from multiple points and tracing the intersection of plumb line paths.