A pendulum clock, comprising a light steel rod attached to a heavy ball, indicates the correct time at 20^ C in a location where g = 9.800 m s ⁻² . If the clock is moved to a new location where g = 9.788 m s ⁻² , at what temperature will it indicate the correct time? The coefficient of linear expansion for steel is 12 10⁻⁶ ^ C ⁻¹ .

Step-by-step solution

The time period of a pendulum is given by T = 2 L g . For the clock to indicate the correct time at the new location, its time period must remain constant, which implies: L₁ g₁ = L₂ g₂ L₂ L₁ = g₂ g₁ Using the formula for linear expansion, L₂ = L₁(1 + ) , where = ₂ - ₁ : 1 + = g₂ g₁ = g₂ - g₁ g₁ Substituting the given values g₁ = 9.800 m s ⁻² , g₂ = 9.788 m s ⁻² , and = 12 10⁻⁶ ^ C ⁻¹ : 12 10⁻⁶ = 9.788 - 9.800 9.800 12 10⁻⁶ = -0.012 9.800 = -0.012 9.800 12 10⁻⁶ = -1000 9.8 -102^ C Since = ₂ - ₁ and ₁ = 20^ C : ₂ - 2