Dimensional Analysis: Simple Pendulum Time Period Solved Question

Dimensional analysis is a useful Class 11 Physics tool for checking equations, finding the dimensions of physical quantities, and testing whether a proposed relation is dimensionally consistent. This original exam-style solved question explains the method step by step.

Question

The time period T of a simple pendulum depends on its length l and acceleration due to gravity g. Using dimensional analysis, obtain a relation of the form

T = k l^a g^b

where k is a dimensionless constant. Determine the values of a and b.

Step 1: Write the dimensions

• Time period: [T] = T
• Length: [l] = L
• Acceleration due to gravity: [g] = LT⁻²

Step 2: Substitute the dimensions

From T = k l^a g^b, and because k has no dimensions,

T = L^a(LT⁻²)^b

Therefore,

T = L^a+bT^−2b

Step 3: Compare the powers

On the left side, the dimension is T¹, with no power of L. On the right side, the powers are L^a+b and T^−2b.

• Comparing the power of L: a + b = 0
• Comparing the power of T: −2b = 1, so b = −1/2

Using a + b = 0, we get a = 1/2.

Final answer

Substitute a = 1/2 and b = −1/2:

T = k l^1/2 g^−1/2 = k√(l/g)

This shows that the time period is proportional to the square root of length and inversely proportional to the square root of gravitational acceleration. Dimensional analysis does not determine the numerical value of k; for an ideal simple pendulum undergoing small oscillations, the full result is T = 2π√(l/g).

Common mistakes to avoid

• Forgetting that acceleration has dimensions LT⁻², not LT⁻¹.
• Comparing dimensions before expanding the powers correctly.
• Assuming dimensional analysis can determine dimensionless constants such as 2π.

Quick practice

Question: Check whether the equation v = u + at is dimensionally consistent, where v and u are velocities, a is acceleration, and t is time.

Key takeaway

To solve dimensional-analysis questions: write the dimensional formula of every quantity, substitute into the proposed relation, simplify the powers, and compare the exponents of each fundamental dimension.

Related Class 11 Physics topics

• Motion: formulas, graphs and solved questions
• Work and Energy
• Explore more Physics notes and solved questions

Note: This is an independently written practice question in an exam-style format, not a verbatim reproduction of an examination paper.