Calorie in New Unit System | Class 11 Physics Dimensional Analysis Solution

 NCERT Class 11 Physics: Calorie Conversion Using α β γ Unit System

-  Dr.Sanjaykumar Pawar 

Step-by-step dimensional analysis showing how calorie is converted into a new unit system using α, β, and γ base units.

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Class 11 Physics (CBSE)

NCERT Exercise 1.3

Question

A calorie is a unit of heat (energy in transit) and it equals about 4.2 J, where

1 J = 1 kg m² s⁻²

Suppose we employ a system of units in which the unit of mass equals α kg, the unit of length equals β m, and the unit of time equals γ s. Show that a calorie has a magnitude 4.2 α⁻¹ β⁻² γ² in terms of the new units.

Topper's Answer

Given:

1 calorie = 4.2 J

1 J = 1 kg m² s⁻²

New units are:

• Unit of mass = α kg
• Unit of length = β m
• Unit of time = γ s

Step 1: Dimensions of Energy

The dimensional formula of energy is:

[M L² T⁻²]

Therefore, one new unit of energy is:

(α kg)(β m)²(γ s)⁻²

= αβ²γ⁻² kg m² s⁻²

= αβ²γ⁻² J

Hence,

1 New Energy Unit = αβ²γ⁻² J

Step 2: Express 1 Joule in New Units

1 J = α⁻¹β⁻²γ² (New Energy Unit)

Step 3: Express 1 Calorie in New Units

1 calorie = 4.2 J

= 4.2(α⁻¹β⁻²γ²)(New Energy Unit)

Therefore,

1 calorie = 4.2 α⁻¹ β⁻² γ²

Hence Proved.

Short Notes (Exam Revision)

• Energy has dimensions [ML²T⁻²].
• New energy unit = αβ²γ⁻² J.
• 1 J = α⁻¹β⁻²γ² (new energy units).
• 1 calorie = 4.2 J.
• Therefore, 1 calorie = 4.2 α⁻¹ β⁻² γ² in the new system of units.

Final Answer: 4.2 α⁻¹ β⁻² γ²

Class 11 Physics

Unit Conversion Practice Questions (Easy to High Level)

Q1 (Easy)

In a system, unit of mass = α kg, unit of length = β m, unit of time = γ s. Find the magnitude of 1 Joule in the new system.

1 J = kg m² s⁻²

Answer:
New unit of energy = αβ²γ⁻² J
So,
1 J = α⁻¹β⁻²γ²

Q2 (Easy)

Find 1 Newton in a system where base units are α kg, β m, γ s.

1 N = kg m s⁻²

Answer:
New form = αβγ⁻² N unit
So,
1 N = α⁻¹β⁻¹γ²

Q3 (Moderate)

Find the magnitude of 1 Pascal in the new system.

1 Pa = kg m⁻¹ s⁻²

Answer:
Substitute new units:
= αβ⁻¹γ⁻²

So,
1 Pa = α⁻¹β⁻³γ²

Q4 (Moderate)

If mass = 2 kg, length = 3 m, time = 5 s, find 1 Joule in new system.

1 J = kg m² s⁻²

Answer:
= 2⁻¹ × 3⁻² × 5²
= 25 / (2 × 9)
= 25/18

Q5 (High)

Show that gravitational constant G has magnitude α⁻¹β³γ⁻² in new system.

G = kg⁻¹ m³ s⁻²

Answer:
Substitute units:
= α⁻¹β³γ⁻²

Q6 (High Concept)

If mass = 4 kg, length = 2 m, time = 1 s, find ratio of new energy unit to Joule.

Energy = M L² T⁻²

Answer:
= 4 × 2² × 1⁻²
= 4 × 4 = 16

Final Answer: 16