Comprehensive theory, key formulas, diagrams, and memory aids for Force and Laws of Motion.
In the previous chapter, we described the motion of an object along a straight line in terms of its position, velocity, and acceleration. But we did not ask: what causes motion? Why does the speed of an object change with time? Do all motions require a cause?
For many centuries, the problem of motion and its causes had puzzled scientists and philosophers. It was believed that the natural state of an object is rest. This belief remained intact until Galileo Galilei and Isaac Newton developed an entirely different approach to understanding motion.
To move a stationary object, or to stop a moving object, we need to push, pull, or hit it. The concept of force is based on this push, pull, or hit. No one has seen, tasted, or felt a force. However, we always see or feel the effect of a force. A force can be used to change the magnitude of velocity of an object (make it move faster or slower) or to change its direction of motion. Force can also change the shape and size of objects (e.g., stretching a rubber band).
Note on Friction: When we push an object on the floor, the floor exerts a force in the opposite direction called friction. To move the object, our pushing force must be greater than the frictional force (resulting in an unbalanced force).
Galileo deduced that objects move with a constant speed when no force acts on them. He observed marbles rolling down inclined planes and reasoned that if the planes were perfectly smooth (zero friction), a marble rolling down one plane would roll up the other to exactly the same height. If the second plane were horizontal, it would roll forever.
This inherent property of all bodies to resist a change in their state of rest or uniform motion is called inertia. * Inertia depends on mass. A heavier object has more inertia than a lighter one. (It's easier to push an empty box than a box full of books). Therefore, mass is a measure of the inertia of a body.
Newton further studied Galileo's ideas and presented three fundamental laws that govern the motion of objects.
The First Law of Motion states: An object remains in a state of rest or of uniform motion in a straight line unless compelled to change that state by an applied unbalanced force.
Because this law defines inertia, it is also known as the Law of Inertia. * Example 1: When a bus suddenly starts moving forward, passengers tend to fall backward. This happens because the lower part of the body moves with the bus, while the upper part tends to remain at rest due to inertia. * Example 2: When a carpet is beaten with a stick, dust comes out of it. The carpet moves forward, but the dust particles try to remain at rest due to inertia and fall down.
The first law tells us that an unbalanced force causes a change in velocity (an acceleration). But how is the acceleration related to the force?
Before defining the second law, we must understand a quantity called momentum (denoted by $p$), introduced by Newton. Momentum is the product of the mass ($m$) and velocity ($v$) of an object. * $p = m \times v$ * Momentum is a vector quantity (has both magnitude and direction). * Its SI unit is kilogram-metre per second ($\text{kg} \cdot \text{m/s}$).
A fast-moving cricket ball or a heavy, slow-moving truck can cause severe injury because they both have high momentum. The force required to stop a moving body depends on its mass and its velocity (its momentum).
The Second Law of Motion states: The rate of change of momentum of an object is proportional to the applied unbalanced force in the direction of force.
Mathematically, if an object of mass $m$ is moving with initial velocity $u$ and is uniformly accelerated to velocity $v$ in time $t$ by a constant force $F$: * Initial momentum, $p_1 = mu$ * Final momentum, $p_2 = mv$ * Change in momentum $= mv - mu = m(v - u)$ * Rate of change of momentum $= \frac{m(v - u)}{t}$
According to the second law: $$F \propto \frac{m(v - u)}{t}$$ Since $\frac{v-u}{t}$ is acceleration ($a$), we get: $$F = k m a$$ (where $k$ is a constant of proportionality). By defining 1 unit of force as the amount that produces an acceleration of $1 \text{ m/s}^2$ in a mass of $1 \text{ kg}$, $k$ becomes 1.
Therefore, the equation is: $$F = m \times a$$
A cricket fielder pulls his hands backwards while catching a fast-moving cricket ball. By doing this, the fielder increases the time ($t$) during which the high velocity of the ball decreases to zero. This decreases the rate of change of momentum, and therefore, the force ($F$) exerted by the ball on the fielder's hands is reduced, preventing injury.
The first two laws tell us how an applied force changes the motion and provide us with a method of determining the force. The third law discusses what happens when two objects interact.
The Third Law of Motion states: To every action, there is an equal and opposite reaction.
Even though the action and reaction forces are always equal in magnitude, these forces may not produce accelerations of equal magnitudes. This is because each force acts on a different object that may have a different mass ($a = F/m$).
Example 1: When a gun is fired, it exerts a forward force (action) on the bullet. The bullet exerts an equal and opposite reaction force on the gun. This results in the recoil of the gun. Because the gun has a much larger mass than the bullet, its acceleration (recoil) is much less than the bullet's acceleration.
Consider two objects (e.g., two balls A and B) colliding. According to the law of conservation of momentum: The total momentum of the two objects before the collision is equal to their total momentum after the collision, provided there is no external unbalanced force acting on them.
If mass of A is $m_A$, initial velocity is $u_A$, and final velocity is $v_A$. If mass of B is $m_B$, initial velocity is $u_B$, and final velocity is $v_B$.
The law can be stated mathematically as: $$m_A u_A + m_B u_B = m_A v_A + m_B v_B$$
This principle is widely used in physics to solve problems involving collisions, explosions, and rocket propulsion.
The study of force allows us to understand the true causes behind the motion described in earlier chapters. Galileo's insights into inertia were formalized by Newton into three laws. The First Law establishes that an unbalanced force is required to change motion. The Second Law provides a mathematical relationship ($F=ma$) linking force to the rate of change of momentum. The Third Law explains that forces always exist in pairs (action and reaction). Finally, the Law of Conservation of Momentum gives us a powerful tool to predict the outcomes of interactions between objects.