the period of a simple pendulum(T) depends on a mass of the bulb(M) length of the tred(L) and acceleration due to gravity(g) use the method to find the expression for the of oscillation(T).
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Delscohtech
To derive the expression for the period of oscillation (T) of a simple pendulum, we can use dimensional analysis.
Step 1: Identify the variables involved in the problem:
- T: period of oscillation (dependent variable)
- M: mass of the bulb (independent variable)
- L: length of the thread (independent variable)
- g: acceleration due to gravity (independent variable)
Step 2: Determine the dimensions of the variables:
- T: time (T)
- M: mass (M)
- L: length (L)
- g: acceleration due to gravity (L/T²)
Step 3: Express the relationship between the variables using powers of the dimensions:
- T ∝ ML⁰(gL/T²)⁰
Step 4: Simplify the expression:
- T ∝ M⁰L⁰g⁰ / L⁰T²⁰
- T ∝ g⁰ / T²⁰
- T ∝ 1 / T²⁰
Step 5: Remove the proportionality sign and introduce a constant of proportionality (k):
- T = k / T²⁰
Step 6: Determine the value of the exponent (n) by equating the dimensions on both sides:
- n = 2⁰
- n = 0
Step 7: Rewrite the equation with the final expression for the period of oscillation (T):
- T = k / T⁰
- T₁ = k / T₂⁰
Therefore, the expression for the period of oscillation (T) of a simple pendulum is:
- T = k / T²⁰
Note: The exact mathematical relationship between these variables may involve a constant value and additional factors that may need to be experimentally determined. However, this derivation provides an expression based on the dimensions of the quantities involved.