Question

3: A simple pendulum with a period of 2 s is used in a clock. The “negligible” bar connecting the pendulum bob to the pivot is made of aluminum (α = 2.4x10^-5/◦C)The clock was built to keep accurate time at 24 ◦C. If the clock is used at an average temperature of 35 ◦C, what correction is needed after 30 days? It will be useful to keep every digit you can throughout the calculation since the differences are quite small. I used 9.81 m/s2 for the acceleration due to gravity.

Answer is 338s. Show me all the work please!

Answer 1

Coefficient of linear expansion = 2.4 x10 ^-5 / ^o C

Period T = 2 s

Length of the pendulam L = L

We know T = 2[L/g]

T ² = 4² (L/g)

From this L = g [T ²/4² )

= 9.81 [2 ² /(4x² ) ]

= 9.81 /²

= 0.9939 m

Length at 35 ^o C is L ' = L [1+(t '- t ) ]

= 0.9930 [1+(2.4x10 ^-5)(35-24)]

= 0.99422 m

Period at 35 ^o C is T '= 2[L ' / g ]

= 2(22/7) [0.99422/9.81]

= 2.0002639 s

30 days = 30 x 24 hrs = 30 x24 x 3600 s

Number of oscillations at 24 ^o C in 30 days , N = (30 x24 x3600) / 2 s

Correction = N(2.0002639 -2)

= 342.12 s