How do you derive the dimensional formula of linear momentum?
Dimensions of Linear Momentum is a core physics topic tested in JEE Main. Recognising and applying the correct dimensional formula ensures accuracy in derivations, units checking, and solving advanced numericals. Understanding how the dimensions relate to physical quantities like mass and velocity is essential for confident problem-solving.
Linear momentum refers to the product of a body's mass and its velocity. It indicates how much motion an object has, and is central to conservation laws and laws of motion. In physics, linear momentum (symbol p) is a vector, always pointing in the direction of velocity.
Dimensional analysis helps students verify equations and convert units in exam settings. The dimensional formula of linear momentum is used to check consistency in both theoretical derivations and real-world applications like collisions, rocket launches, and impulse calculations.
Definition and Physical Meaning of Dimensions of Linear Momentum
The dimensions of linear momentum express how it depends on fundamental units: mass (M), length (L), and time (T). Momentum is given by p = m × v, where m is mass and v is velocity.
In SI, mass is measured in kilograms and velocity in metres per second. Linear momentum then has units of kg·m/s. Dimensional analysis expresses this relationship as a product of the dimensions of mass and velocity, a common approach in units and dimensions topics.
Deriving the Dimensional Formula of Linear Momentum
- Start with the formula: p = m × v.
- Mass (m): dimension is [M].
- Velocity (v): dimension is [L T-1].
- Multiply: [M] × [L T-1] = [M1 L1 T-1].
- So, the dimensional formula of linear momentum is [M1 L1 T-1].
This step-by-step method can be used for other derived quantities such as force, impulse, and torque. Practising these derivations deepens your understanding of momentum and related physics dimensions.
Units and SI Dimensions of Linear Momentum
| Physical Quantity | SI Unit | CGS Unit | Dimensional Formula |
|---|---|---|---|
| Linear Momentum | kg·m/s | g·cm/s | [M1L1T-1] |
Understanding SI and CGS units is crucial for tackling units and measurements conversion questions in JEE. The dimensions of linear momentum also allow you to validate solutions in kinematics, dynamics, and collision problems.
Sample JEE Main Example using Dimensions of Linear Momentum
A body of mass 5 kg moves at a velocity of 8 m/s. Find its linear momentum, and check the dimensional consistency of the result.
- p = m × v = 5 × 8 = 40 kg·m/s
- kg·m/s has the dimension [M1 L1 T-1]
- The formula and answer are dimensionally correct.
JEE questions frequently require this type of direct calculation or equation validation using the dimensional formula of linear momentum.
- Many collision and conservation of momentum problems use this approach.
- Impulse (change in momentum) uses the same dimensions — see the impulse dimensional formula.
- Checking units helps avoid errors in laws of motion problem sets.
Comparison: Momentum vs Related Quantities
| Quantity | Dimensional Formula |
|---|---|
| Linear Momentum | [M1L1T-1] |
| Impulse | [M1L1T-1] |
| Force | [M1L1T-2] |
| Angular Momentum | [M1L2T-1] |
| Torque | [M1L2T-2] |
Notice the contrast between linear momentum and torque or angular momentum. These differences prevent common errors when matching formulas to physical meaning on test day.
- Momentum and impulse share dimensions, but represent different concepts.
- Force has an extra negative exponent for time, as indicated by Newton’s second law.
- Angular momentum’s extra power of L is a standard JEE pitfall.
Common Mistakes and Quick Tips for Dimensions of Linear Momentum
- Confusing the formula: Remember p = m × v for linear, not angular momentum.
- Dropping the negative exponent "-1" for time in the dimension [T-1].
- Mixing units: Stick to SI (kg, m, s) in all JEE calculations unless asked.
- Misreading exam questions: Distinguish between force, impulse, and momentum using their dimensional formulas.
- Always check: Is the answer’s unit or dimension consistent? This avoids careless mistakes.
Always use dimensional analysis to confirm answer plausibility—especially under exam pressure. Dimensional reasoning is also vital when applying techniques learned in topics like units and dimensions, laws of motion, and measurement.
In summary, the dimensions of linear momentum are fundamental for verifying physics equations and tackling a wide range of JEE Main problems. Regularly derive them from first principles to strengthen your exam performance. For more advanced applications, see the contrast with moment of inertia or surface tension.
- For further revision, browse physics formulas and solved question pages on Vedantu.
- Test yourself on applications by using kinematics mock tests or laws of motion questions.
- For deeper context, cross-check your answers with dimensions of work or experiment with kinematics concepts.
All definitions, derivations, and concepts here align with the latest JEE Main syllabus and NCERT standards. The explanations are crafted by physics educators, and deeper topics on Vedantu expand your learning journey.
FAQs on Dimensions of Linear Momentum: Definition, Formula & Applications
1. What is the dimensional formula of linear momentum?
Linear momentum has the dimensional formula MLT-1.
• Here, M stands for mass (kg), L is length (m), and T is time (s).
• The formula is derived as: Linear momentum = mass × velocity = [M] × [LT-1] = MLT-1.
• This is an important dimension used in exams and physics syllabus for class 11 and competitive exams.
2. What is the dimensional formula of angular momentum?
Angular momentum has the dimensional formula ML2T-1.
• It is obtained as angular momentum = moment of inertia × angular velocity = [ML2] × [T-1] = ML2T-1.
• This dimensional formula differentiates it clearly from linear momentum.
• It is important in both rotational motion chapters and numerical questions.
3. What is the dimension of momentum in physics?
The dimension of momentum in physics is MLT-1.
• This applies to both the SI and CGS systems.
• Mass contributes M, velocity provides LT-1.
• Understanding this helps in dimensional analysis and checking equation validity.
4. What is the dimensional formula of the moment of inertia?
Moment of inertia has the dimensional formula ML2.
• Calculated as moment of inertia = mass × (distance)2 = [M][L]2 = ML2.
• It represents rotational inertia in physics and is covered in class 11 syllabus under rotational motion.
5. What is the dimensional formula of impulse?
Impulse shares the same dimensional formula as linear momentum, which is MLT-1.
• Impulse = Force × time = [MLT-2] × [T] = MLT-1.
• This is important for numericals and conceptual questions in physics exams.
6. What is the formula for linear momentum?
The linear momentum formula is p = m × v.
• Here, p = linear momentum, m = mass, v = velocity.
• SI unit is kg·m/s.
• Linear momentum is a vector quantity and plays a key role in mechanics.
7. What is the dimensional formula of surface tension?
Surface tension has the dimensional formula MT-2.
• Calculated as force per unit length: [MLT-2] / [L] = MT-2.
• Important for fluids chapter and exam preparation in class 11 physics.
8. Are the dimensions of linear momentum and impulse the same?
Yes, linear momentum and impulse have the same dimensional formula: MLT-1.
• Both relate to mass, velocity, force, and time.
• This similarity is often asked in competitive and class 11 board exams.
9. How are the dimensions of torque related to those of linear momentum?
Torque has the dimensional formula ML2T-2, while linear momentum has MLT-1.
• Torque derives from force × distance = [MLT-2] × [L] = ML2T-2.
• Unlike linear momentum, torque shows an extra power of L and T-1, linking it to rotational motion.
10. Why is it important to know the dimensional formula of linear momentum?
Knowing the dimensional formula helps verify equations, convert units, and solve numerical problems.
• It aids in understanding relationships between physical quantities.
• Useful for identifying errors in equations.
• Essential for dimensional analysis and physics practical exams.
11. Is the dimensional formula of linear momentum and angular momentum the same?
No, the dimensional formula for linear momentum (MLT-1) is different from angular momentum (ML2T-1).
• The extra L in angular momentum accounts for rotational motion.
• Understanding their difference is crucial for exam and conceptual clarity.
12. What is the SI unit of linear momentum?
The SI unit of linear momentum is kilogram metre per second (kg·m/s).
• This unit directly reflects its dimensional formula MLT-1.
• Both numerical and theoretical questions require this knowledge in exams.
13. How do you find the dimensions of linear momentum?
To find the dimensions of linear momentum, multiply mass and velocity using their base dimensions:
1. Mass (m): [M]
2. Velocity (v): [LT-1]
3. Linear momentum (p = m × v): [M][LT-1] = MLT-1
• This method ensures clarity for board and NEET/JEE aspirants.
