In the previous lessons we discussed motion(position,velocity,acceleration) along a straight line with uniform and non-uniform motion, now we will discuss what causes motion.
Causes of Motion
The concept of force is based on push, hit, or pull. It changes the magnitude, direction , shape, and size of an object.
Balanced and Unbalanced Forces
Balanced forces: Equal forces that do not cause a change in the state of motion.
Unbalanced forces: Forces that cause an object to move in the direction of the greater force.
Friction: A force that opposes motion, acting between surfaces in contact.
An object maintains its motion when forces are balanced, resulting in no net external force.
An unbalanced force is required to accelerate an object, changing its speed or direction.
First Law of Motion
Galileo deduced that objects move at a constant speed when no force acts on them.
An unbalanced force is needed to change motion, but no force is needed to sustain uniform motion.
Newton's First Law of Motion: An object remains at rest or in uniform motion unless acted upon by an external force.
Inertia: The tendency of objects to resist changes in their state of motion.
The First Law of Motion is also known as the Law of Inertia.
Safety belts in cars are used to counteract inertia and prevent accidents during sudden stops.
Inertia causes a person to fall backwards when a bus starts suddenly and to lean to one side when a car turns sharply.
Activity 8.1
A pile of carom coins demonstrates inertia. When the bottom coin is hit, the inertia of the other coins causes them to fall vertically.
Galileo Galilei (1564 – 1642)
Galileo Galilei was born on February 15, 1564, in Pisa, Italy.
Galileo had an early interest in mathematics and natural philosophy.
In 1586, he wrote ‘The Little Balance [La Balancitta]’, describing Archimedes’ method for finding relative densities.
In 1589, he presented theories about falling objects using inclined planes in ‘De Motu’.
In 1592, he became a mathematics professor at the University of Padua.
Galileo formulated the law for uniformly accelerated objects, noting that distance is proportional to the square of time.
He developed improved telescopes and designed the first pendulum clock around 1640.
Galileo’s astronomical discoveries, including mountains on the moon and the Milky Way's composition, were published in ‘Starry Messenger’.
He argued that planets orbit the Sun, contrary to popular belief, based on observations of Saturn and Venus.
Activity 8.2
A five-rupee coin on a card over a glass tumbler falls into the tumbler when the card is flicked away due to inertia.
Activity 8.3
A water-filled tumbler on a tray spills when turned around quickly due to inertia.
Inertia and Mass
Inertia: the resistance offered by an object to change its state of motion.
Mass: A measure of an object's inertia.
Heavier objects have larger inertia.
Second Law of Motion
The first law of motion indicates that when an unbalanced external force acts on an object, its velocity changes, that is, the object gets an acceleration.
The impact produced by objects depends on their mass and velocity.
Momentum p of an object is defined as the product of its mass m and velocity v.
p = mv (8.1)
Momentum has both direction and magnitude, with its direction being the same as that of velocity.
The SI unit of momentum is kilogram-meter per second \text{kg m s}^{-1}.
A force produces a change of momentum.
The change of momentum of an object depends on the time during which the force is exerted.
The force necessary to change the momentum of an object depends on the time rate at which the momentum is changed.
The second law of motion states that the rate of change of momentum of an object is proportional to the applied unbalanced force in the direction of force.
Mathematical Formulation of Second Law of Motion
Consider an object of mass m moving along a straight line with an initial velocity u.
It 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: \propto p2 – p1 \propto mv – mu \propto m × (v – u)
Rate of change of momentum: \propto m \frac{v – u}{t}
Applied force: F \propto m \frac{v – u}{t}
F = k m \frac{v – u}{t} (8.2)
F = kma (8.3)
a = \frac{v – u}{t} is the acceleration.
k is a constant of proportionality.
The SI units of mass and acceleration are kg and m s-2 respectively.
The unit of force is chosen such that k = 1.
1 unit of force = k × (1 \text{ kg}) × (1 \text{ m s}^{-2}).
From Eq. (8.3): F = ma (8.4)
The unit of force is kg m s-2 or newton (N).
The second law provides a method to measure force as the product of mass and acceleration.
Increasing the time during which a force is applied reduces the impact, as seen when a fielder pulls their hands back while catching a cricket ball.
In high jump, athletes land on cushioned or sand beds to increase the time of impact and reduce the force.
From the mathematical expression for the second law of motion, the first law of motion can be mathematically stated:
F = ma
F = m \frac{v - u}{t} (8.5)
Ft = mv – mu
When F = 0, v = u for any time t, indicating uniform motion, or rest if u = 0.
Example 8.1
A constant force acts on a 5 kg object for 2 s, increasing its velocity from 3 m/s to 7 m/s. Find the force magnitude.
If the force is applied for 5 s, calculate the final velocity.
Solution:
Given:
u = 3 \text{ m s}^{-1}
v = 7 \text{ m s}^{-1}
t = 2 \text{ s}
m = 5 \text{ kg}
Using Eq. (8.5): F = m \frac{v - u}{t}
F = 5 \text{ kg} \frac{(7 \text{ m s}^{-1} – 3 \text{ m s}^{-1})}{2 \text{ s}} = 10 \text{ N}
For t = 5 \text{ s}, the final velocity is: v = u + \frac{Ft}{m}
v = 3 \text{ m s}^{-1} + \frac{(10 \text{ N})(5 \text{ s})}{5 \text{ kg}} = 13 \text{ m s}^{-1}
Example 8.2
Which requires greater force: accelerating a 2 kg mass at 5 m/s² or a 4 kg mass at 2 m/s²?
Solution:
Using F = ma
For m1 = 2 \text{ kg}, a1 = 5 \text{ m s}^{-2}: F1 = m1 a_1 = 2 \text{ kg} × 5 \text{ m s}^{-2} = 10 \text{ N}
For m2 = 4 \text{ kg}, a2 = 2 \text{ m s}^{-2}: F2 = m2 a_2 = 4 \text{ kg} × 2 \text{ m s}^{-2} = 8 \text{ N}
Conclusion: F1 > F2, so accelerating the 2 kg mass requires greater force.
Example 8.3
A motorcar moving at 108 km/h stops in 4 s after applying brakes. Calculate the brake force if the car's mass with passengers is 1000 kg.
Solution:
Initial velocity u = 108 \text{ km/h} = 30 \text{ m s}^{-1}
Final velocity v = 0 \text{ m s}^{-1}
Mass m = 1000 \text{ kg}
Time t = 4 \text{ s}
Using F = m \frac{v - u}{t}
F = 1000 \text{ kg} × \frac{(0 - 30) \text{ m s}^{-1}}{4 \text{ s}} = -7500 \text{ N}
The negative sign indicates the force opposes the motion.
Example 8.4
A force of 5 N gives mass m1 an acceleration of 10 m/s² and mass m2 an acceleration of 20 m/s². What acceleration would it give if both masses were tied together?
Combined mass m = m1 + m2 = 0.50 \text{ kg} + 0.25 \text{ kg} = 0.75 \text{ kg}
Acceleration of combined mass a = \frac{F}{m} = \frac{5 \text{ N}}{0.75 \text{ kg}} = 6.67 \text{ m s}^{-2}
Example 8.5
A 20 g ball moves along a straight line on a table, and its velocity decreases to zero in 10 s due to friction. How much force does the table exert on the ball?
Solution:
Given:
Initial velocity, u = 20 \text{ cm s}^{-1} = 0.2 \text{ m s}^{-1}
Final velocity, v = 0 \text{ cm s}^{-1} = 0 \text{ m s}^{-1}
Time, t = 10 \text{ s}
Mass, m = 20 \text{ g} = 0.02 \text{ kg}
Acceleration, a = \frac{v-u}{t} = \frac{0 - 0.2 \text{ m s}^{-1}}{10 \text{ s}} = -0.02 \text{ m s}^{-2}
Force, F = ma = 0.02 \text{ kg} × (-0.02 \text{ m s}^{-2}) = -0.0004 \text{ N}
The negative sign indicates that the force is frictional and acts opposite to the ball's motion.
Third Law of Motion
The first two laws of motion describe how applied forces change motion and how to measure force.
The third law states that when one object exerts a force on another, the second object exerts an equal and opposite force back on the first.
These forces act on different objects and never on the same object.
Colliding with another player in football illustrates this law; both players experience a pair of forces.
These opposing forces are known as action and reaction forces.
An alternative statement: to every action, there is an equal and opposite reaction.
Action and reaction always act on two different objects simultaneously.
Even though action and reaction forces are equal in magnitude, they may not produce equal accelerations due to different masses.
When a gun is fired, the forward force on the bullet results in an equal recoil of the gun. The gun's acceleration is less due to its greater mass.
A sailor jumping out of a boat demonstrates this law: the sailor moves forward, and the boat moves backward.
Activity 8.4
Two children on separate carts throw a sandbag to each other.
Each child experiences an instantaneous force when throwing the bag.
The motion of the carts can be observed by painting a white line on the wheels.