Pendulum of unknown length and unknown g, only period given

[mh1985]

Homework Statement

Pendulum of unknown length, l, has period of 9.32 seconds. Length is increased by 1 metre, and time period increases to 9.734 seconds. Calculate original length of pendulum and determine whether the experiment was conducted on earth. Hint: g is not assumed to be 9.81 ms^-2, you must find both g & l

Homework Equations

x(t) = A1 cos v_n^t + A2 sin v_n^t?

The Attempt at a Solution

Really not sure where to start, due to the two unknowns!

 

[mfb]

Can you express the period as function of g and l?
For both pendulum lengths, you get an equation, so you have two unknowns and two equations.

 

[mh1985]

Can you express the period as function of g and l?
For both pendulum lengths, you get an equation, so you have two unknowns and two equations.

T=2pi(sqrt(l/g) ?

 

[mfb]

That is right, and it will help you.

 

[Nickie]

As a scientist, it is important to use scientific methods and principles to solve problems. In this case, we can use the equation for the period of a pendulum, T = 2π√(l/g), where T is the period, l is the length of the pendulum, and g is the acceleration due to gravity. We also know that the period, T, is directly proportional to the length, l, and inversely proportional to the acceleration due to gravity, g.

Using this information, we can set up two equations:

T = 2π√(l/g) and T' = 2π√((l+1)/g)

Where T is the original period (9.32 seconds) and T' is the new period (9.734 seconds).

We can then solve for g in both equations:

g = 4π²l/T² and g = 4π²(l+1)/T'²

Since both equations are equal to g, we can set them equal to each other and solve for l:

4π²l/T² = 4π²(l+1)/T'²

Simplifying, we get:

l/T² = (l+1)/T'²

Cross-multiplying, we get:

lT'² = (l+1)T²

Expanding, we get:

lT'² = lT² + T²

Subtracting lT² from both sides, we get:

lT'² - lT² = T²

Factoring out l, we get:

l(T'² - T²) = T²

Dividing both sides by (T'² - T²), we get:

l = T²/(T'² - T²)

Plugging in the given values for T and T', we get:

l = (9.32 seconds)²/((9.734 seconds)² - (9.32 seconds)²)

Solving this, we get:

l = 38.1 meters

Therefore, the original length of the pendulum is 38.1 meters. To determine if the experiment was conducted on Earth, we can compare the calculated value of g with the known value of g on Earth, which is approximately 9.81 m/s². If the calculated value of g is close to 9.81 m/s²,