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Chapter 1: Problem 123

Dimensional formula for calories is .............. (a) \(\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-2}\) (b) \(\mathrm{M}^{2} \mathrm{~L}^{1} \mathrm{~T}^{-2}\) (c) \(\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}\) (d) \(\mathrm{M}^{2} \mathrm{~L}^{2} \mathrm{~T}^{-2}\)

Short Answer

Expert verified
The short answer to the question is: (c) \(\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}\).

Step by step solution

01

Understanding Dimensional Analysis

Dimensional analysis is a method used to determine the relationship between different physical quantities by identifying their dimensions (such as mass, length, time, etc.). The dimensional formula of a physical quantity is the expression relating that quantity to the base quantities in terms of dimensions.
02

Understanding Calories

A calorie is a unit of energy. More specifically, it is defined as the amount of energy required to raise the temperature of 1 gram of water by 1 degree Celsius. To find the dimensional formula for calories, we need to understand it as an energy unit.
03

Relating Calories to Energy

Energy can be understood in terms of the mechanical work done, which is represented by the following formula: \(W = F \cdot d\) where \(W\) is work (energy), \(F\) is force, and \(d\) is displacement.
04

Determine the Dimensional Formula for Force

Force can be expressed using Newton's second law of motion: \(F = m \cdot a\) where \(F\) is force, \(m\) is mass, and \(a\) is acceleration. The dimensions of mass are: \([m] = \mathrm{M}\) The dimensions of acceleration are: \([a] = \dfrac{\mathrm{L}}{\mathrm{T^2}}\) So, the dimensions of force are calculated as: \([F] = \mathrm{M} \cdot \dfrac{\mathrm{L}}{\mathrm{T^2}} = \mathrm{M^1 L^1 T^{-2}}\)
05

Determine the Dimensional Formula for Displacement

Displacement is a change in position and has the same dimensions as length. Therefore, the dimensions of displacement are: \([d] = \mathrm{L}\)
06

Determine the Dimensional Formula for Calories (Energy)

As energy (\(E\)) is equal to work (\(W\)), we'll find the dimensions of work by substituting the dimensions of force and displacement: \([W] = [F] \cdot [d] = \mathrm{M^1 L^1 T^{-2}} \cdot \mathrm{L} = \mathrm{M^1 L^2 T^{-2}}\) Thus, the dimensional formula for calories (energy) is \(\mathrm{M^1 L^2 T^{-2}}\). Comparing with the given options, the answer is (c) \(\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}\).

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Most popular questions from this chapter

If the centripetal force is of the form $\mathrm{m}^{\mathrm{a}} \mathrm{v} \mathrm{b}_{\mathrm{r}}^{\mathrm{c}}$ find the values of \(\mathrm{a}, \mathrm{b}\) and \(\mathrm{c}\) (a) \(1,2,1\) (b) \(1,2,-1\) \(\begin{array}{ll}\text { (c) } 1,3,-2 & \text { (d) }-1,3,-1\end{array}\)

The period of oscillation of a simple pendulum is given by $\mathrm{T}=2 \pi \sqrt{(} \ell / \mathrm{g})$ what is the equation of the relative error \(\Delta \mathrm{T} / \mathrm{T}\) in measurement of period \(\mathrm{T}\) ? (a) \((1 / 2)(\Delta \ell / \ell)\) (b) \(2(\Delta \ell / \ell)\) (c) \((1 / 4)(\Delta \ell / \ell)\) (d) \(4(\Delta \ell / \ell)\)

The unit of Stefen Boltzman constant \((\sigma)\) is ............ (a) \(\mathrm{w}^{2} \mathrm{~m}^{-2} \mathrm{k}^{-1}\) (b) \(\mathrm{w} \mathrm{m}^{2} \mathrm{k}^{-3}\) (c) \(\mathrm{w} \mathrm{m}^{-2} \mathrm{k}^{4}\) (d) \(\mathrm{w} \mathrm{m}^{-2} \mathrm{k}^{-4}\)

The force \(F\) is represented by equation \(F=P \ell+Q \ell^{-1}\), where \(\ell\) is the length. The unit of \(P\) is same as that of $\ldots \ldots \ldots \ldots \ldots$ (a) Surface tension (b) velocity (c) force (d) momentum

Equation of \(\ell_{1}=\ell_{0}\left[1+\alpha\left(\mathrm{T}_{2}-\mathrm{T}_{1}\right)\right]\) find out the dimensions of the coefficient of linear expansion \(\alpha\) suffix. (a) \(\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{1} \mathrm{~K}^{1}\) (b) \(\mathrm{M}^{0} \mathrm{~L}^{1} \mathrm{~T}^{1} \mathrm{~K}^{1}\) (c) \(\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{0} \mathrm{~K}^{1}\) (d) \(\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0} \mathrm{~K}^{-1}\)
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