A pendulum clock can be approximated as a simple pendulum of length 1.03 m and keeps accurate time at a location where g = 9.82 m/s2. In a location where g = 9.77 m/s2, what must be the new length of the pendulum, such that the clock continues to keep accurate time (that is, its period remains the same)?
T = k sqrt L/g
sqrt(1.03/9.82) = sqrt (L/9.77)
so
L = 9.77 (1.03/9.82)
Why did the pendulum clock go on a diet? Because it wanted to keep accurate time even in locations with different values of g!
To find the new length of the pendulum, we can use the formula for the period of a simple 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.
Since we want the period to remain the same, we can equate the two formulas for the period in the two locations:
2π√(1.03/9.82) = 2π√(new length/9.77)
By simplifying the equation, we get:
√(1.03/9.82) = √(new length/9.77)
Squaring both sides of the equation, we get:
1.03/9.82 = new length/9.77
Cross multiplying, we get:
new length = (1.03/9.82) * 9.77
Calculating the value, the new length of the pendulum should be approximately 1.0265 meters. So, the pendulum needs to lose just a little bit of length to keep accurate time in the new location with g = 9.77 m/s^2.
To find the new length of the pendulum, we can use the formula for the period of a simple pendulum:
T = 2π√(L/g)
Where:
T is the period
L is the length of the pendulum
g is the acceleration due to gravity
In this case, we need to find the new length L', to keep the period the same. So, we have:
T = 2π√(1.03/9.82)
We can rewrite this equation as:
T = 2π√(L'/9.77)
Since the period T remains the same, we can equate the two equations:
2π√(1.03/9.82) = 2π√(L'/9.77)
Cancelling out the 2π on both sides, we get:
√(1.03/9.82) = √(L'/9.77)
Squaring both sides, we have:
1.03/9.82 = L'/9.77
Simplifying, we find:
L' = (1.03/9.82) * 9.77
Calculating this, we get:
L' ≈ 1.0306 m
Therefore, the new length of the pendulum should be approximately 1.0306 meters.
To find the new length of the pendulum that will keep accurate time in a location with a different value of gravitational acceleration, we need to use the formula for the period of a simple pendulum:
T = 2π * √(L / g)
where:
T = period of the pendulum
L = length of the pendulum
g = acceleration due to gravity
Given that the original length of the pendulum is 1.03 m and the original value of g is 9.82 m/s^2, we can calculate the original period of the pendulum using the given formula.
T1 = 2π * √(1.03 / 9.82)
Now, we need to find the new length of the pendulum for the clock to continue keeping accurate time in a location where g = 9.77 m/s^2. Let's call this new length L2.
The condition for the clock to keep accurate time is that the period of the pendulum remains the same. Therefore, we can set T1 equal to the new period, which will be calculated using the new length (L2) and the new value of g (9.77 m/s^2).
T1 = T2 = 2π * √(L2 / 9.77)
Now, we can equate the two expressions for T1 and T2:
2π * √(1.03 / 9.82) = 2π * √(L2 / 9.77)
We can simplify this equation by canceling out the 2π on both sides:
√(1.03 / 9.82) = √(L2 / 9.77)
Now, let's isolate L2 by squaring both sides of the equation:
1.03 / 9.82 = L2 / 9.77
Next, cross-multiply to solve for L2:
L2 = (1.03 / 9.82) * 9.77
L2 ≈ 1.028 m
Therefore, the new length of the pendulum should be approximately 1.028 meters in the location where g = 9.77 m/s^2 in order for the clock to continue keeping accurate time.