Newton’s Third Law of Motion states that forces always occur in pairs. When one object exerts a force on another, the second object simultaneously exerts an equal force in the opposite direction on the first object.
Statement:- "For every action, there is an equal and opposite reaction."
Mathematically,
\boxed {\overrightarrow{\rm F}_{AB} = -\overrightarrow{\rm F}_{BA}}
where FAB is the force exerted by body A on body B, and FBA is the force exerted by body B on body A.
Action and Reaction Pairs in Nature
- Ball and Wall: When a ball hits a wall, the ball pushes the wall, and the wall pushes the ball back with equal force, causing it to rebound.
- Walking: While walking, the foot pushes the ground backward; the ground pushes the foot forward.
- Swimming: A swimmer pushes water backward, and water pushes the swimmer forward.
- Birds Flying: Birds push air downward; air pushes the birds upward, providing lift.
- Gravity: Earth pulls an object downward, and the object pulls Earth upward with the same force.
Key Features of Action (Reaction Forces)
- Equal in magnitude
- Opposite in direction
- Act on different bodies
- Occur simultaneously
- Do not cancel each other, since they act on separate objects
Proof that Action and Reaction are Equal and Opposite
To understand the concept of the action and reaction forces, let us consider a system of two spring balances, A and B, connected.
The spring balance B is fixed to any rigid support. A force is then applied to the loose and free end by pulling the spring balance A. As an effect of the application of force, both spring balances show the same reading.
This concludes that both spring balances witness equal-magnitude forces. It also shows that the force exerted by both spring balances, A on B, is equal but opposite in direction to the force exerted by spring balance B on A.
Here, the force exerted on the acting body (the spring balance A on B) is termed as "action," and the force exerted by the reacting body (spring balance B on A) can be termed as "reaction."
Mathematical Interpretation
Action and reaction pair states that every action has equal and opposite reaction. For a system of two bodies, A and B, let us assume FAB is a force of body An action on B and FBA is a force by B on body A.
The mathematical expression w.r.t. the forces is given by,
FAB = - FBA
where,
- FAB is an action on B
- FBA is reaction of body B on A
Negative sign indicates that the force acting on body A is in the opposite direction to the force that is acting on body B.
Derivation of Law of Action and Reaction
Derivation of Law of Action and Reaction from Newton's Second Law of Motion is:
Let us assume an isolated system with no external forces acting upon it, consisting of two massive bodies, A & B, mutually interacting with each other. Now, Let us consider both bodies to be in effect of a force under the influence of each other, that is, FAB, to be the force exerted on body B by body A and FBA be the force exerted by body B on A.
Due to these forces, FAB and FBA, let us assume dp1/dt and dp2/dt to be the rates of the change of momentum in the effect of these bodies, respectively. Then,
FBA = dp1/dt and FAB = dp2/dt
Adding these equations is done as follows:
FBA + FAB = dp1/dt + dp2/dt
FBA + FAB = d(p1 + p2) / dt
Since no external force acts on the system, therefore,
d(p1 + p2) / dt = 0
or
FBA + FAB = 0
FBA = -FAB
The above equation represents Newton's third law of motion (i.e., for every action there is an equal and opposite reaction).
Solved Problems
Problem 1: A car with a mass of 1250 Kg traveling with an acceleration of 10 m/s2 hits a bike. What force does the car experience?
Solution:
given , Mass(m) = 1250kg , acceleration(a) = 10m/s2
Using Newton's second law F = ma:
F = 1250 x 10 = 12500 N
the force experienced by the car due to hitting the bike would be equal in magnitude and opposite in direction to the force experienced by the bike due to being hit by the car. However, the magnitude of the force experienced by the car is not equal to the acceleration; it's equal to the mass multiplied by the acceleration. Hence, the car experiences a force of 12500 N.
Problem 2: A Dog of mass 10 kg jumps on a table of mass 60 kg. As the Dog walks around on the table, what is the average force that the table applies to the Dog? Use g = 10 m/s2.
Solution: The force that the dog applies to the table is its weight. As per Newton's third law, the table also applies a force to the dog of the same magnitude.
The force on the dog from the table is:
Fs = FN = ma = 10 kg × 10 m/s2 = 100 N
Problem 3: A boy is riding his scooter and pushes off the ground with his foot. Thus, this causes him to accelerate at a rate of 8 m/s2. Boy's weight is 600 N. What is the strength of his push off the ground? Use g = 10 m/s2.
Solution:
Boy's weight, F is 600 N.
The formula to calculate the force on an object is,
F = ma
where m is the mass and a is the acceleration.
or
m = F / a
m = 600 N / 10 m/s2
m = 60 kg
Boy accelerates at 8 m/s2. so, he is pushed by a force of
F = ma = 60 kg × 8 m/s2 = 480 N
Problem 4: Two bodies apply forces to each other. The force on one of the bodies as a function of time in the x-direction is kt + b, where k and b are constants. What's the force as a function of time in the x-direction on the other body? Consider that no other forces are present besides the forces the bodies apply to each other.
Solution:
According to Newton's third law of motion, the force exerted by one body on another is equal in magnitude and opposite in direction to the force exerted by the second body on the first. Therefore, if the force on one body in the x-direction is kt + b
then the force on the other body in the x-direction would be -kt-b This ensures that the forces between the two bodies satisfy Newton's third law.