Problem 1 on Units & Dimensions

Description:

Problem

A calorie is a unit of heat or energy and it equals 4.2J where, 1J = 1 kg m²s². Suppose we employ a system of units in which the unit of mass equals α kg, the unit of length equals β m, the unit of time is γ s. Show that a calorie has a magnitude 4.2α¹β²γ² in terms of the new units.

Solution

Understanding the concept of changing base units,

Old Standards	New Standards

Taking example of Length first (First Row) −

• Old standards define 1m as the fundamental unit of length. (YELLOW – First Column)

• Therefore, the red rod is measured to be 5m. (RED – First Column)

• New standards define 2m as the fundamental unit of length. (YELLOW – Second Column)

• Therefore, the same red rod is measured to be 2.5m. (RED – Second Column)

Therefore, we can say,

New Measurement = (old Measurement) × (Old Unit/New Unit)

Same goes for the Mass (Second Row) −

• Old standards define 1kg as the fundamental unit of length. (YELLOW – First Column)

• Therefore, the red block is measured to be 5kg. (RED – First Column)

• New standards define 2kg as the fundamental unit of length. (YELLOW – Second Column)

• Therefore, the same red rod is measured to be 2.5kg. (RED – Second Column)

Therefore, we can again say,

New Measurement = (old Measurement) × (Old Unit/New Unit)

Note − This is not a formula to be remembered by the students. This can be intuitively understood.

Coming to the problem, it is given,

1 J = 1 kg m²s^-2

New fundamental units defined,

New Unit Mass = (α) kg, Hence, Old Unit Mass = 1/α(New Unit Mass)

New Unit Length = (β) m, Hence, Old Unit Length = 1/β(New Unit Length)

New Unit Time = (γ) s, Hence, Old Unit Time = 1/γ(New Unit Time)

Now,

1 calorie = 4.2 J

1 calorie = 4.2 kg m²s^-2

1 cal = 4.2 (1 old unit mass)(1 old unit length)²(1 old unit time)^-2

1 calorie = 4.2 (1/α kg)(1/β m)²(1/γ s)^-2

1 calorie = 4.2 α^-1β²γ^-2 kg m²s^-2