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Force and Acceleration are related to each other according to Newton’s Second law of motion.
- Newton's laws are among the most fundamental laws of physics.
- According to the first law, a body will remain at rest or in uniform motion until a net external force acts on it.
- According to the second law, the rate of change of linear momentum of a body is directly proportional to the force applied.
- In other words, we can say that force applied to a body is the product of mass and acceleration of the body.
- Force is an interaction that causes a body to change its condition, such as from rest to motion or vice versa.
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Key Terms: Force, Acceleration, Velocity, Motion, Momentum, Dimensional formula, Unit of force, Unit of acceleration, Newton’s Law, Newton’s second of motion, Uniform Motion
Definition of Force and Acceleration
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The definition of force and acceleration is given below
Force
Force is an external force that can change the condition of rest or motion of a body. It is also described as the push or pull on an object with a particular mass that causes it to change velocity.
- Whenever time two objects interact, a force is applied to each of them.
- Both objects no longer feel the force when their interaction is ended.
- Forces exist only as a result of interactions.
A force can affect a body in three different ways.
- It can change the speed of the body.
- It can change the direction of the motion.
- It can change the shape of the body.
Forces acting on a body can be divided into two groups
- Contact Force: Contact forces occur when two interacting objects are directly in contact with one another. Contact forces include friction, tension, normal, air resistance, and applied forces.
- Non-contact Force: Non-contact forces occur even when two interacting objects are not physically in contact with each other. Examples of non-contact forces are Gravitation force, Electrostatic force, and Magnetic force.
Acceleration
Acceleration refers to the rate at which velocity changes over time. Any change in the velocity of the object causes acceleration.
- If the velocity of the moving body increases with time, then it is referred to as positive acceleration.
- If the velocity of the moving body decreases with time, then it is referred to as negative acceleration, deceleration, or retardation.
- If a body is moving in a circular path with a constant speed, then the acceleration of the body is known as centripetal acceleration.
- Newton's second law states that acceleration is directly proportional to the sum of all forces acting on an object, and inversely proportional to its mass.
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Units of Force and Acceleration
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Force and acceleration are vector quantities. The direction in which force is applied is referred to as the direction of the force, and the point at which force is applied is referred to as the application of force.
Unit of Force
The SI unit of force is Newton (N). The dimensional formula of force is [F] = [M L T-2].
One newton is defined as the force required to accelerate one kilogram of mass at the rate of one meter per second squared.
1 N = 1 Kg m/s2
Unit of Acceleration
The SI unit of acceleration is the meter per second squared (m/s2). The dimensional formula of acceleration is [a] = [M0 L T-2].
Some formulas for Equations of motion under constant acceleration are given below
- v = u + at
- v2 = u2 + 2aS
- S = ut + 1/2 at2
Where
- S is the distance traveled by the body
- u and v are the initial and final velocities
- t is the time
Relationship Between Force and Acceleration
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The relationship between force and acceleration is given by Newton’s second law of motion. According to this law, the rate of change of linear momentum of a body is directly proportional to the applied force. Therefore, we can write
F ∝ ∆p/∆t
⇒ F = k ∆p/∆t
Where
- k is the constant of proportionality.
- ∆p is the change in momentum
- ∆t is the change in time
Taking ∆t → 0, the term ∆p/∆t becomes dp/dt.
⇒ F = k dp/dt
Also, momentum, p = m x v
⇒ F = k d(mv)/dt
Since mass m is constant, therefore we can write
F = km (dv/dt)
According to the definition of acceleration, it is the rate of change in velocity, therefore
acceleration, a = dv/dt
⇒ F = k (ma)
By taking k = 1, we get
F = ma
Therefore, the relationship between force and acceleration is given by
F = ma
Where
- F is the force
- m is the mass
- a is the acceleration
This relationship shows that acceleration is determined not only by the force applied to it but also by the mass of the body to which it is applied. When the same force is applied to two different bodies, the one with a lighter mass experiences more acceleration.
- We also see that the larger the force, the greater the acceleration.
- Hence, there must be a relationship between force and acceleration.
- This is what Newton's Second Law states.
Things to Remember
- Force and Acceleration are related to each other according to Newton’s Second law of motion.
- Force is an external force that can change the condition of rest or motion of a body.
- Acceleration refers to the rate at which velocity changes over time.
- The SI unit of force is Newton (N).
- The SI unit of acceleration is the meter per second squared (m/s2).
- The larger the force, the greater the acceleration.
Sample Questions
Ques. What is force? (2 Marks)
Ans. Force is an external force that can change the condition of rest or motion of a body. It is also described as the push or pull on an object with a particular mass that causes it to change velocity.
Ques. What is acceleration? (1 Mark)
Ans. Acceleration refers to the rate at which velocity changes over time. Any change in the velocity of the object causes acceleration.
Ques. It is observed that passengers are thrown forward from their seats when a speeding bus stops suddenly, why? (2 Marks)
Ans. When the speeding bus stops suddenly, the lower part of the body in contact with the seat comes to rest but the upper part of the body of the passengers tends to maintain its uniform motion. Hence the passengers are thrown forward.
Ques. A person of mass 70 kg stands on a weighing scale on a lift. If the lift is descending with a downward acceleration of 9 m/s2, what would be the reading of the weighing scale? Use g = 10 m / s2 (3 Marks)
Ans. Let R be the reading on the weighing scale, then,
Or, ( 70 g ) − R = 70 × 9
Or, R = 700 − 630 = 70N
So the reading scale will display 7 kg
Ques. The velocity of a body if given as a function of t is - v = 2 t i + t 2 j
If the mass of the body is 2 kg, then find the momentum and the force acting on it, at time t = 2 s (3 Marks)
Ans. Momentum is given by, p = M v
And F = (d / dt) p
Or, F = M (d / dt) ( 2 t i + t 2 j )
Or, F = 4 i +4 t j
Therefore at 2 s force F = 4 i + 8 j
Ques. State Newton’s Second law and give its mathematical expression (3 Marks)
Ans. Statement. It states the time rate of change of momentum of a body is directly proportional to the external force applied to it and the change in momentum takes place in the direction of the force.
Mathematically, the Force ∝ rate of change of momentum
While momentum p = M v (by definition)
We get force F = (d / dt) p
Or, F = (d / dt) (M v)
Or, F = M (d / dt) v
Or, F = M a
Ques. Cricketers likely lower their hands while catching a ball. Why? (2 Marks)
Ans. The impulse received by the hand, while holding a catch is equal to the product of the force applied and the time taken to complete the catch. By moving his hands backward, a cricketer increases the time for completing the catch. As a result, the force that acts on his hands becomes much smaller and does not hurt him.
Ques. Why is it desired to hold a gun tight to one’s shoulder when it is being fired? (2 Marks)
Ans. Since the gun recoils after firing, it must be held lightly against the shoulder because the gun and the shoulder constitute one system of greater mass so the backkick will be less.
Ques. On a smooth horizontal surface, two blocks of masses M and m are connected by a light spring. The two blocks are pulled and then released. Show that their acceleration ratio is inversely proportional to their masses. (3 Marks)
Ans. The forces F M and F m due to masses M and m act in opposite directions. Thus, we get,
F M + F m = 0
Or, M a M + m a m = 0
Or, M a M = - m a m
Or, a M / a m = - m / M ( hence proved )
Ques. A force is applied to a body, but it produces no acceleration. How do you explain the observation? (3 Marks)
Ans. If a force is being applied to a body but it causes no acceleration then two possibilities may arise -
- If the force is a deforming force then it does not produce acceleration.
- The force is an internal force that cannot cause acceleration.
Ques. Two concurrent forces of 16N and 12N are acting on a mass of 200kg in mutually perpendicular directions. What is the magnitude of the acceleration produced? (3 Marks)
Ans. F = ( 16 2 + 122 + 2 cos 90)½
Or, F = ( 16 2 + 122 )½
Or, F = 20 N
Or, a = F / m = 20 / 200 = 0.1m / s2
Therefore the magnitude of acceleration produced is 0.1 m / s2
Ques. Write the consequences of Newton’s second law of motion. (3 Marks)
Ans. Newton’s second law of motion has several consequences some of them are -
- Concept of inertial mass.
- A body can be accelerated only through a force.
- The measurement of applied force
- The amount of acceleration produced by a force depends only on the mass of the body.
- The applied force can be measured from the force applied to the mechanism.
Ques. A monkey of mass 40 kg climbs on a rope that can stand a maximum tension of 600 N. In which of the cases will the rope break, if
(a) climbs up and has an acceleration of 6 m / s2
(b) climbs down with an acceleration of 4 m / s2
(c) climbs up and has a uniform speed of 5 m/s
(d) freely falls from the rope under gravity.
Given, g = 10 m / s2 and ignore the mass of the rope. (4 Marks)
Ans.
The monkey of mass 40 kg, T = 600 N (max tension rope can hold)
The rope will break if reaction (R) exceeds Tension (T)
(a) a = 6 m / s2
Or, R = m ( g + a ) = 40 ( 10 + 6 ) = 640 N (Rope will break)
(b) a = 4 m / s2
Or, R = m (g – a) = 40 (10 – 6) = 240 N (Rope will not break)
(c) uniform speed of v = 5 m/s (constant) a = 0
Or, R = mg = 40 10 = 400 N (Rope will not break)
(d) a = g; R = m ( g – a ) = m ( g – g )
R = zero ( Rope will not break)
Ques. A force of F = - k x acts on a particle of mass 0.5 kg. What will be its initial acceleration if it is released from a point 50 cm away from the origin if k = 30 N m -1? (3 Marks)
Ans. F = - k x, given k = 30 N m -1 . x = 50 cm = 0.5m
Or, F = - 30 0.5 N = - 15 N
Or, a = F / M = - 15 / 0.5
Or, a = - 30 m / s2
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