How to Derive the Dimensional Formula for Force Constant Easily
The dimension of force constant is an important fundamental concept in physics, particularly within the study of oscillations, elasticity, and mechanical systems. Understanding its dimensional formula is essential for both theoretical analysis and numerical problem-solving in competitive exams like JEE Main.
Definition of Force Constant
The force constant, commonly denoted as $k$, quantifies the stiffness of a spring or elastic system. It indicates the amount of force required to produce a unit displacement in the system, as described by Hooke’s Law. This property plays a critical role in analyzing the behavior of vibrating systems and oscillatory motion.
Physical Significance of Force Constant
The magnitude of the force constant determines how resistant a spring is to deformation. A higher value of $k$ denotes a stiffer spring which requires more force to stretch or compress by a certain length, while a lower value signifies a more easily deformable spring.
Dimensional Formula of Force Constant
To derive the dimensional formula of the force constant, consider Hooke’s Law, which relates the applied force $F$ to the displacement $x$ through the equation $F = -kx$. The force constant $k$ can be represented as $k = \dfrac{F}{x}$.
The dimensional formula for force is $[M^1L^1T^{-2}]$ and for displacement is $[L^1]$. Substituting these in:
$k = \dfrac{[M^1 L^1 T^{-2}]}{[L^1]} = [M^1 L^0 T^{-2}]$
Thus, the dimensional formula of force constant is $[M^1L^0T^{-2}]$.
Derivation Using Dimensional Analysis
Force can be expressed as the product of mass and acceleration, where acceleration has the dimensions $[L^1 T^{-2}]$. Hence, force is represented by $[M^1 L^1 T^{-2}]$. Displacement retains the dimension $[L^1]$.
Applying the formula $k = \dfrac{F}{x}$ gives:
$[k] = \dfrac{[M^1 L^1 T^{-2}]}{[L^1]} = [M^1 L^{1-1} T^{-2}] = [M^1 L^0 T^{-2}]$
This confirms that the force constant is dimensionally dependent on mass and time, but independent of length.
SI Unit of Force Constant
The SI unit of the force constant is newton per metre (N/m). This derives directly from the definition $k = \dfrac{F}{x}$, where force is measured in newtons and displacement in metres. The higher the value of N/m, the stiffer the material or spring.
| Quantity | SI Unit |
|---|---|
| Force Constant ($k$) | $\text{N m}^{-1}$ |
| Force ($F$) | $\text{N}$ |
| Displacement ($x$) | $\text{m}$ |
Summary of Dimensional Formula and Units
The dimension of force constant is $[M^1L^0T^{-2}]$, which means it depends on mass and the square of inverse time. Its SI unit is newton per metre (N/m).
Applications of Force Constant
Force constant is significant in the study of oscillations, determining frequencies of vibrating systems, analysing mechanical shocks, and evaluating the properties of elastic materials. These concepts are foundational for solving problems involving simple harmonic motion and mechanical resonance.
Examples and Practice
A spring with a force constant of $100\,\text{N/m}$ requires a force of $50\,\text{N}$ to extend by $0.5\,\text{m}$, as $F = kx$. Understanding the relationship between force, displacement, and stiffness is necessary for application-based problems in examinations.
Related Dimensional Formulas in Physics
Other physical quantities, such as the Dimensions Of Stress, also play a key role in mechanics and elasticity. These relationships are based on dimensional analysis similar to that used for the force constant.
The dimensional approach for force constant finds parallels in the study of physical quantities including Dimensions Of Resistance, Dimensions Of Speed, and Dimensions Of Magnetic Flux.
Comprehensive understanding of dimensional formulas supports systematic problem-solving across areas such as Dimensions Of Density and Dimensions Of Volume for advanced physics study.
FAQs on What Is the Dimension of Force Constant in Physics?
1. What is the dimension of the force constant?
The dimension of the force constant (also called spring constant) is the set of fundamental physical quantities required to express it, commonly in terms of mass (M), length (L), and time (T).
- Force constant is defined as force per unit length (κ = F/x).
- Expressed dimensionally: [F] / [L]
- Since force F = [M L T-2], we get:
- [Force constant] = [M L T-2] / [L] = [M T-2]
This is a key term in physics, especially in SHM and elasticity topics.
2. What is the SI unit and dimensional formula of force constant?
The SI unit of force constant is newton per meter (N/m), and its dimensional formula is [M T-2].
- SI unit: N/m
- Dimensional formula: [M T-2]
These relate to mass and time in physics and mechanics.
3. Why is the dimensional formula for force constant [M T-2]?
The force constant is defined as force per unit extension (F/x), so its dimensional formula is [M T-2].
- Force (F) = [M L T-2]
- Displacement (x) = [L]
- Thus, [k] = [F]/[x] = [M L T-2] / [L] = [M T-2]
This formula highlights its role in elasticity and simple harmonic motion.
4. What is the physical significance of the force constant?
The force constant measures the stiffness of a spring or elastic object; a higher value indicates a stiffer spring.
- Represents resistance to deformation
- Determines oscillation frequency in simple harmonic motion (SHM)
- Key factor in Hooke’s Law
This is a fundamental concept in physics and engineering.
5. What is the dimensional formula of Hooke's law constant (k)?
The dimensional formula of Hooke’s law constant (k) is [M T-2].
- Hooke’s law: F = -kx
- k = F/x
- [k] = [M L T-2] / [L] = [M T-2]
This applies to the force constant in spring systems.
6. Which quantities have the same dimension as the force constant?
Quantities that have the same dimension as the force constant [M T-2] include:
- Tension (when divided by length)
- Surface tension (force per unit length)
- Elastic modulus (with appropriate scaling)
This is useful for dimensional analysis and problem solving.
7. Is the force constant a scalar or vector quantity?
The force constant is a scalar quantity. It has magnitude only and no specific direction.
- Represents stiffness regardless of force direction
- Used in magnitude calculations in Hooke’s Law
This makes it easier to use in numerical and theoretical problems.
8. What is the difference between force constant and spring constant?
The terms 'force constant' and 'spring constant' are often used interchangeably in physics.
- Both refer to the same physical quantity (k)
- Both have the same dimensional formula: [M T-2]
- Both measure the stiffness of a spring or elastic object
Some sources may use 'force constant' in a broader sense for elastic materials in general.
9. How do you derive the dimensional formula of force constant?
To derive the dimensional formula of force constant (k):
- Start with Hooke’s Law: F = -kx
- Rearrange: k = F/x
- Force (F) has dimension: [M L T-2]
- Displacement (x) has dimension: [L]
- So, [k] = [M L T-2]/[L] = [M T-2]
This stepwise method is useful for exams and assignments.
10. What is the dimensional analysis importance of force constant in physics?
Dimensional analysis of force constant helps in checking the correctness of equations, comparing quantities, and converting units.
- Ensures physical consistency in derivations
- Helps verify Hooke’s law and SHM formulas
- Assists in identifying equivalent units (e.g., N/m or kg s-2)
This is important in CBSE exams and competitive entrance tests.
