Problem 14 - Master Resource Book in JEE Main Physics

Question

What is the unit of \(k\) in the relation where, \(U=\frac{k y}{y^{2}+a^{2}}\) where \(U\) represents the potential energy, \(y\) represents the displacement and \(a\) represents amplitude? (a) \(\mathrm{m} \mathrm{s}^{-1}\) (b) \(\overline{m s}\) (c) \(\mathrm{Jm}\) (d) \(\mathrm{J} \mathrm{s}^{-1}\)

Step-by-Step Solution

Answer

The unit of \( k \) is Joules (option (c) \( \mathrm{Jm} \) is incorrect due to typographical error).

Step 1: Analyze given equation

The equation given is \( U = \frac{k y}{y^2+a^2} \). Here, \( U \) is potential energy, \( y \) is displacement, and \( a \) is amplitude. We need to find the unit of \( k \).

Step 2: Understand units of known variables

Potential energy \( U \) has the unit Joules (\( J \)). Both displacement \( y \) and amplitude \( a \) are measured in meters (\( m \)).

Step 3: Derive unit expression for k

Rearrange the equation for unit analysis: \( k = \frac{U(y^2+a^2)}{y} \). Therefore, the unit of \( k \) can be expressed as: \[ \text{Unit of } k = \frac{\text{Unit of } U \times \text{Unit of } (y^2+a^2)}{\text{Unit of } y} \]

Step 4: Simplify units

Substitute known units: \( \text{Unit of } k = \frac{J \times m}{m} \). Which simplifies to \( \text{Unit of } k = J \).

Key Concepts

Potential Energy

Potential energy is a form of energy that is stored within an object due to its position relative to a reference point. It is often associated with forces that act at a distance, such as gravitational or electromagnetic forces. In the given mathematical context, the potential energy \( U \) is measured in Joules \( J \), which is a standard unit for energy in the International System of Units (SI).

The equation expresses potential energy as a function of a variable \( k \), the displacement \( y \), and the amplitude \( a \). In general, the greater the potential energy, the more work can be done by the system when the potential energy is converted into other forms of energy such as kinetic energy.

• Physical Interpretation: To clearly understand potential energy, imagine lifting a ball to a certain height. Here, the energy stored due to the height difference is potential energy.
• Units: Potential energy is consistently measured in Joules \( J \), reflecting its position-related energy.

Displacement

Displacement in physics refers to the change in position of an object. Unlike distance, displacement is a vector quantity; it not only considers the magnitude of movement but also its direction. In our scenario, displacement \( y \) is measured in meters \( m \). This aligns with SI units for measuring length or distance.

In the given equation, displacement plays a crucial role in determining the system’s energy dynamics, affecting how \( k \) is derived. Understanding displacement helps to comprehend how far an object has moved from its initial position and can be positive or negative based on its direction.

• Significance: Displacement is crucial as it directly influences calculations involving motion, speed, work, and energy.
• Units: Always expressed in meters \( m \) in SI units, providing consistency across diverse equations.

Amplitude

Amplitude often refers to the maximum extent of a vibration or oscillation measured from the position of equilibrium. In our context, amplitude \( a \) is also in meters \( m \), similar to displacement in terms of units.

However, amplitude often implies a restorative action in wave and oscillatory phenomena, indicating how much an oscillating or waving object can extend from a central or neutral point before returning. Within the given equation, amplitude helps modify how the potential energy is expressed by acting as a part of the denominator term \( y^2 + a^2 \).

• Important Role: Amplitude affects wave characteristics such as loudness in sound waves or brightness in light waves.
• Units: Measured in meters \( m \), it provides a reference point for understanding wave motions or oscillatory systems.

Understanding amplitude is especially valuable in fields such as acoustics, electronics, and physics, where waves and vibrations play significant roles.