Force analysis in accelerated frame

The description, measurement and analysis of motion or force system in inertial and accelerated (non-inertial) frames are different. As the motion of an object in two systems should represent same physical phenomenon, it is desirable that the differences are resolved and made consistent to each other.

Our objective here is to reconcile the differences. For this, we consider a block of mass 10 kg lying on the floor of a lift, which itself is moving up with an acceleration of 2 $m/s^2$ . For the sake of convenience, we consider acceleration due to gravity is equal to 10 $m/s^2$ . Here, we have taken numerical values so that we have the feel of the difference of measurements in inertial and accelerated reference systems.

Inertial frame of reference (ground)

An observer on the ground, who can see the block (let us think that the lift is covered with transparent glass), makes following measurements/ assumptions based on his/her understanding of classical mechanics :

• The block is moving up with an acceleration, a = 2 $m/s^2$ .
• The acceleration due to gravity is 10 $m/s^2$ . The acceleration due to gravity in the vicinity of earth surface is a constant and should not change because of the acceleration of the lift.
• The weight of the block is mg = 10 x 10 = 100 N.
• The block is acted upon by a normal force, N, as applied by the floor of the lift in upwards direction.

He/ She analyzes forces, applying Newton’s second law of motion. The free body diagram of the block is :

$$\begin{array}{l}\sum F_y=N-mg=m{a}_y=ma\\ \Rightarrow N=m(g+a)=10(10+2)=120\mathrm{Newton}\end{array}$$

The observer, therefore, concludes (applying Newton’s third law) that the force, which the block exerts on the floor is equal in magnitude, N = 120 N as against 100 N, when the lift would have been stationary.

He/ She anticipates that a spring balance will register a value of 120 N as it measures normal force applied on it. We must know that spring balance measures normal force - not the weight. Clearly, the measurement is not same as the weight of the block, mg (10 x10 = 100 N), when lift in stationary. The normal force is equal to weight only when spring balance is stationary in inertial frame.

Accelerated frame of reference (lift)

As against above, an observer in the lift makes following measurements/ assumptions based on his/her understanding of classical mechanics :

• The block is at rest.
• The acceleration due to gravity is 10 $m/s^2$ (This does not change in the vicinity of Earth).
• The weight of the block is mg = 10 x 10 = 100 N. This is based on the fact that mass of the block is 10 kg. Hence, weight, mg = 10 X 10 = 100 N.

He analyzes the force applying Newton’s second law of motion. The free body diagram of the block is :

$$\begin{array}{l}\sum F_y=N-mg=0\\ \Rightarrow N=mg=10x10=100\mathrm{Newton}\end{array}$$

This observer is not capable to perceive and hence measure the acceleration of the lift. He, however, can measure the normal force on the block, say with a spring balance. He/ She find that spring balance actually measures it to be 120 N! Clearly, there is something with his force analysis. The basic difference in the analysis of two observers arises due to the fact that observer on the ground recognizes block as an accelerated body, whereas observer on the lift recognizes block as stationary.