Units and Dimensions - Parallax

Description:

Concept of Parallax

The change in the apparent position of an object because of the change in the point of observation is called parallax.

Distance between the two points of observation is called basis.

In the above figure,

• First observation is made from View 1. The object appears to be present with the blue screen in its background.

• Second observation is made from View 2. The object appears to be present with the red screen in its background.

This phenomenon of varying apparent position of the object happening simply by changing the position of observation is called parallax.

Measuring large distances

In the above figure,

• ‘S’ is the far-away point (at a large distance).

• ‘A’ and ‘B’ are the points of observation.

• ‘d’ is the distance of S from ‘A’ and ‘B’.

• (It is equal for ‘A’ and ‘B’ because ‘S’ is very far from each.)

• ‘b’ is the distance between ‘A’ and ‘B’. (b ≪ d)

• ‘θ’ (in radians) is the parallax angle.

• Goal is to calculate the value of ‘d’.

Because d is very large, the line AB (length b) can be interpreted as the arc of a very large circle (of radius d). Now using the definition of plane angle

θ (in radians) = b/d

This can be re-written as −

d = b/θ (in radians)

Hence, the value of ‘d’ can be calculated.

Angle Measurement in Physics

Units used for measurement of angles are −

• Radians
• Degrees
• Minutes
• Seconds

Following are the conversions among them −

• 2π rad = 360^o

• 1 rad = 180^o/π

• 1^o = π/180 rad

Now,

• 1′ = (1/60)^o, 1 minute is defined as one-sixtieth of a degree.

• 1′′ = (1/60)′, 1 second is defined as one-sixtieth of a minute.

Example on Parallax problem

Problem − The moon is observed from two diametrically opposite points A and B on earth. The angle θ subtended at the moon by the two directions of observation is 1^o54′. Given the diameter of earth to be about 1.276 × 10⁷ m, compute the distance of the moon from the earth.

Solution −

Given in the problem,

• θ = 1^o54′ = (114/60)^o = (0.0105 π) rad

• Angle has to be put in terms of radians. Degree values are not accepted.

• b = 1.276 × 10⁷ m

To be calculated, d.

Using the formula, d = b/θ, d = 1.276 × 10⁷/0.0105 × π

Hence, d = 3.86 × 10⁸ m - Answer