How to Calculate the Dimensions of Displacement with Examples
The concept of displacement and its dimensional representation is fundamental in physics, particularly in the study of kinematics. Displacement refers to the change in position of an object and is characterized as a vector quantity. Understanding the dimensions of displacement is essential for analyzing various physical quantities and their interrelations in classical mechanics.
Definition and Characteristics of Displacement
Displacement is defined as the shortest distance between the initial and final positions of an object in a specific direction. It is a vector quantity, having both magnitude and direction, and is measured using a straight line joining the two points. Displacement is different from distance, which is a scalar and represents the total path length traveled irrespective of direction.
Dimensional Formula of Displacement
Dimensional analysis allows physical quantities to be expressed in terms of fundamental units such as mass (M), length (L), and time (T). As displacement denotes a change in position, it is associated only with length. Therefore, the dimensional formula of displacement is written as $[\mathrm{M}^0 \mathrm{L}^1 \mathrm{T}^0]$.
| Physical Quantity | Dimensional Formula |
|---|---|
| Displacement | $[\mathrm{M}^0 \mathrm{L}^1 \mathrm{T}^0]$ |
| Velocity | $[\mathrm{M}^0 \mathrm{L}^1 \mathrm{T}^{-1}]$ |
| Distance | $[\mathrm{M}^0 \mathrm{L}^1 \mathrm{T}^0]$ |
Physical Meaning and Unit of Displacement
The displacement of an object measures its net change in position from the starting point to the endpoint. The SI unit of displacement is the meter (m), identical to the unit of length. No units of mass or time are present in its dimensional expression.
Comparison of Displacement with Related Quantities
Displacement and distance both represent a change in position, yet they differ fundamentally. Displacement is a vector quantity, while distance is a scalar. For a given motion, displacement can be positive, negative, or zero, depending on the direction, whereas distance is always non-negative. Detailed dimensional analysis is explained in related topics, such as Dimensions Of Speed.
Mathematical Expression for Displacement
Displacement can be mathematically represented as the difference between the final and initial positions of an object. For one-dimensional motion, displacement $\Delta x$ is given by:
$\Delta x = x_f - x_i$
where $x_f$ is the final position and $x_i$ is the initial position.
Solved Examples on Displacement
The calculation of displacement in simple cases involves subtracting the initial coordinate from the final coordinate. For instance, if an object starts at 2 m and ends at 8 m, the displacement is:
$\Delta x = 8\,\mathrm{m} - 2\,\mathrm{m} = 6\,\mathrm{m}$
Similarly, if an object starts at 5 m and moves to 1 m, its displacement is:
$\Delta x = 1\,\mathrm{m} - 5\,\mathrm{m} = -4\,\mathrm{m}$
Vector Nature and Representation
Because displacement is a vector, it can be represented graphically by an arrow whose length corresponds to the magnitude and whose direction shows the direction of displacement. Mathematical operations on displacement vectors enable the determination of resultant displacement in cases involving multiple steps or segments of motion.
Applications of Dimensional Analysis to Displacement
Analyzing the dimensions of displacement aids in the verification of physical equations involving kinematic quantities. For example, velocity is defined as displacement divided by time, so its dimensions are $[\mathrm{L}\ \mathrm{T}^{-1}]$. More examples of such analyses can be found in the article on Dimensions Of Work.
Difference Between Displacement and Distance
Distance always refers to the length of the path traveled, while displacement describes the shortest direct distance between starting and ending points. Displacement may be less than or equal to the distance for any motion. Both have the same dimensions, but only displacement is directional.
Key Points on Dimensional Formula of Displacement
- Displacement is a vector quantity
- SI unit is meter (m)
- Dimensional formula: $[\mathrm{M}^0 \mathrm{L}^1 \mathrm{T}^0]$
- Represents net change in position
- Magnitude may be less than or equal to distance
Summary Table: Dimensional Formulae of Common Quantities
| Quantity | Dimensional Formula |
|---|---|
| Displacement | $[\mathrm{M}^0 \mathrm{L}^1 \mathrm{T}^0]$ |
| Distance | $[\mathrm{M}^0 \mathrm{L}^1 \mathrm{T}^0]$ |
| Speed | $[\mathrm{M}^0 \mathrm{L}^1 \mathrm{T}^{-1}]$ |
| Force | $[\mathrm{M}^1 \mathrm{L}^1 \mathrm{T}^{-2}]$ |
Further Reading and Related Topics
A comprehensive understanding of the dimensional formula of displacement supports the study of other physical quantities such as speed, force, work, and volume, each with its own dimensional representation. Additional resources on these topics can be accessed at Dimensions Of Volume and Dimensions Of Density.
FAQs on Understanding the Dimensions of Displacement
1. What are the dimensions of displacement?
Displacement has the same dimensions as length, specifically [L], where L represents length.
Key points:
- Displacement is a vector quantity.
- Its dimension is L1 or simply length.
- In SI units, displacement is measured in metres (m).
2. What is displacement in physics?
Displacement is defined as the shortest distance from the initial to the final position of a point and is a vector quantity.
Key features:
- Shows both magnitude and direction.
- Represents a change in position.
- Mathematically, displacement = final position − initial position.
3. How is displacement different from distance?
Displacement measures the shortest straight-line distance between two points, while distance is the total path length traveled.
Main differences:
- Displacement is a vector; distance is a scalar.
- Displacement can be zero or negative; distance is always positive.
- Units for both are metres (m) in SI.
4. What is the formula for displacement?
The formula for displacement is:
Displacement = Final Position − Initial Position
Details:
- Expressed as Δx = xfinal − xinitial
- For motion in 3D, Δr = rfinal − rinitial
5. Is displacement a scalar or a vector quantity?
Displacement is a vector quantity, which means it has both magnitude and direction.
Key characteristics:
- Indicates direction of motion.
- Can have positive, negative, or zero value depending on direction.
6. What are the units of displacement in the SI system?
In the SI system, the unit of displacement is the metre (m).
Related information:
- For very large distances, kilometres (km) may be used.
- For small distances, centimetres (cm) or millimetres (mm) may also be used.
7. How do you express the dimensions of displacement?
The dimensions of displacement are expressed as [L].
Points to remember:
- No mass (M) or time (T) is involved.
- Dimensionally: [M0 L1 T0]
8. Can displacement be negative?
Yes, displacement can be negative depending on the direction of movement relative to a reference point.
Details:
- Negative displacement indicates movement in the opposite direction.
- Sign depends on the chosen coordinate system.
9. What is the difference between position and displacement?
Position describes the location of a point, while displacement measures the change in position.
Important points:
- Position is an absolute point; displacement is a relative change.
- Displacement = final position − initial position.
10. Why are the dimensions of displacement important?
The dimensions of displacement are important for analyzing motion and checking the consistency of physical equations.
Benefits:
- Helps in dimensional analysis.
- Assists in verifying formulas in kinematics.
- Ensures units are correctly applied in calculations.
