Problem 4 - Master Resource Book in JEE Main Physics

Question

Given that \(r=m^{2} \sin p t\), where \(t\) represents time. If the unit of \(m\) is \(\mathrm{N}\), then the unit of \(r\) is (a) \(N\) (b) \(\mathrm{N}^{2}\) (c) \(\mathrm{N}\)-s (d) \(\mathrm{N}^{2} \mathrm{~s}\)

Step-by-Step Solution

Answer

The unit of \(r\) is \(N^2\).

Step 1: Analyze the Formula

Given the formula \(r = m^2 \sin(pt)\), we need to determine how the units in this expression combine to give the unit of \(r\).

Step 2: Determine the Unit of \(m^2\)

\(m\) is given to have the unit of \(\mathrm{N}\). Therefore, \(m^2\) has the unit: \[ (N)^2 = N^2 \].

Step 3: Determine the Unit of \(\sin(pt)\)

\(\sin(pt)\) is a trigonometric function that is dimensionless. Therefore, it has no unit.

Step 4: Combine Units to Determine the Unit of \(r\)

Since \(m^2\) has a unit of \(N^2\) and \(\sin(pt)\) is dimensionless, the unit of \(r = m^2 \sin(pt)\) is simply \(N^2\).

Key Concepts

Dimensional Analysis

Dimensional analysis is a powerful tool used in physics to determine the relationships between different physical quantities. It helps us check the consistency of equations and can predict how different parameters will affect each other based on their dimensions. When performing dimensional analysis, we look at the fundamental dimensions such as mass (M), length (L), time (T), and sometimes current (I), temperature (Θ), and more.
This method allows physicists to verify the correctness of equations by ensuring that both sides of the equation have the same dimensions. It also aids in converting one set of units to another by using conversion factors. In the provided exercise, dimensional analysis is used to ascertain the correct units of the variable \( r \) based on the units given for \( m \). By examining the dimensional integrity of the equation \( r = m^2 \sin(pt) \), we see that the units of \(r\) are determined by the units of \( m^2 \) since the \(\sin(pt)\) is dimensionless.

Trigonometric Functions in Physics

Trigonometric functions, such as \( \sin \), \( \cos \), and \( \tan \), are frequently used in physics because they relate the angles of triangles to the lengths of their sides, which helps describe periodic phenomena. For instance, waves and oscillations often involve these functions.
In the expression \( r = m^2 \sin(pt) \) given in the original exercise, \( \sin(pt) \) illustrates how trigonometric functions are used to account for periodic behavior without altering the fundamental units of the equation. The sine function, like most trigonometric functions, is dimensionless. This means it does not contribute any units or dimensions to the variables it multiplies or itself.

• Dimensional consistency: Trigonometric arguments, such as angles, must be dimensionless; hence the product \( pt \) leads to \( \sin(pt) \) being dimensionless.
• Application: Useful in analyzing cycles and waveforms, they help in predicting and describing motion and forces that vary with time.

Physical Quantities and Units

Physical quantities in physics are usually represented by a numerical value along with a unit. Units give meaning to the number and allow scientists and engineers to communicate and perform calculations with these quantities effectively. Different physical quantities have specific units depending upon the type of measurement involved.
For example, in the problem provided,

• \( m \), which has the unit of Newtons (N), typically represents a force-related parameter.
• \( r \), resulting from a function involving \( m \), therefore inherits its unit dependency from \( m^2 \).

The correct determination of units is crucial for ensuring equations are set up correctly and that resulting quantities make sense. The unit of any expression can be determined using basic rules of algebra applied to units, similar to how we determine numerical calculations. By identifying that \( \sin(pt) \) is dimensionless and does not carry units itself, we focus solely on \( m^2 \) for determining \( r \)'s units as \( N^2 \). This systematic use of units helps in understanding and applying the relationships between different physical quantities effectively.