If mass is measured in unit of alpha kg,...

If mass is measured in unit of `alpha` kg, length in `beta` m and time in `gamma ` s, then calorie would be

`4 2 alpha beta^(2)gamma^(-2)`

`[MT^(-2)]`

`42alpha^(-1)beta^(-2)gamma^(-2)`

`42alpha^(-2)beta^(-2)gamma^(-2)`

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To determine the unit of calorie when mass is measured in `alpha` kg, length in `beta` m, and time in `gamma` s, we need to start by understanding the relationship between calories and joules, as well as how to express joules in terms of the given units. ### Step-by-Step Solution: 1. **Understand the definition of Joule**: The SI unit of energy is the Joule (J). One Joule is defined as the work done when a force of one newton displaces an object by one meter in the direction of the force. Mathematically, it can be expressed as: \[ 1 \text{ Joule} = 1 \text{ Newton} \times 1 \text{ meter} \] 2. **Express Newton in terms of base units**: A newton (N) is defined as the force required to accelerate a mass of one kilogram at a rate of one meter per second squared. Therefore: \[ 1 \text{ Newton} = 1 \text{ kg} \cdot \text{m/s}^2 \] 3. **Substituting Newton into the Joule equation**: Now substituting the expression for Newton into the Joule definition: \[ 1 \text{ Joule} = 1 \text{ kg} \cdot \text{m/s}^2 \cdot 1 \text{ m} = 1 \text{ kg} \cdot \text{m}^2/\text{s}^2 \] 4. **Convert Joules to calories**: We know that: \[ 1 \text{ calorie} = 4.184 \text{ Joules} \] Therefore, we can express calories in terms of Joules: \[ 1 \text{ calorie} = 4.184 \text{ kg} \cdot \text{m}^2/\text{s}^2 \] 5. **Substituting the given units**: Now, substituting the given units where mass is `alpha` kg, length is `beta` m, and time is `gamma` s: \[ 1 \text{ calorie} = 4.184 \cdot \alpha \cdot \beta^2/\gamma^2 \] 6. **Final expression**: Thus, the unit of calorie in terms of the new units is: \[ \text{calorie} = 4.184 \cdot \alpha \cdot \frac{\beta^2}{\gamma^2} \] ### Summary: The calorie, when mass is measured in `alpha` kg, length in `beta` m, and time in `gamma` s, is given by: \[ \text{calorie} = 4.184 \cdot \alpha \cdot \frac{\beta^2}{\gamma^2} \]

Topper's Solved these Questions

PRACTICE SET 19
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PRACTICE SET 21
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Suppose two students are trying to make a new measurement system so that they can use it like a code measurement. Instead of taking 1 kg, 1 m 1 s, as the basic unit they took unit of mass as alpha kg, the unit of length as beta m and the unit of tme as gamma s. They called the new system as ACME. Column I represents the units of the physical quantities, Column II represents the ACME units and Column III represents the CGS units of the physical quantities. Which of the following represents the correct combination for a physical quantity known as rate of energy?

(III) (iii) (L)

(IV) (ii) (J)

(II) (i) (M)

(I) (iv) (K)

If units of length, mass and force are chosen as fundamental units, the dimensions of time would be :

`M^(1//2) L^(-1//2) F^(1//2)`

`M^(1//2) L^(1//2) F^(1//2)`

`M^(1//2) L^(1//2) F^(-1//2)`

`M^1 L^(-1//2) F^(-1//2)`

If units of length, mass and force are chosen as fundamental units, the dimensions of time would be: