If a pendulum-driven clock gains 5.00 s/day, what fractional change
in pendulum length must be made for it to keep perfect time?
Please provide explanations and solutions. Thanks in advance :D
To determine the fractional change in pendulum length that would allow the clock to keep perfect time, we need to analyze the relationship between the clock's gain in time and the length of the pendulum.
First, let's establish some important concepts:
1. Period: The period of a pendulum is the time it takes for it to complete one full swing. It is typically denoted by the symbol T.
2. Frequency: The frequency of a pendulum indicates how many swings it completes in a given time. It is the reciprocal of the period and is usually denoted by the symbol f.
3. Pendulum Length: The length of the pendulum is the distance between the point of suspension (where the pendulum is connected) and the center of mass. It is typically denoted by the symbol L.
Now, let's consider the relationship between the period and length of a pendulum. According to the mathematical equation for the period of a pendulum:
T = 2π * √(L / g)
where:
- T is the period of the pendulum,
- π (pi) is a constant approximately equal to 3.14159,
- L is the length of the pendulum, and
- g is the acceleration due to gravity.
From this equation, we can see that the period depends on the square root of the pendulum length.
Now, let's focus on the given information. The clock gains 5.00 s per day. To determine the fractional change in pendulum length required, we need to consider the gain in time per swing. Since there are 24 hours in a day, we can calculate the gain in time per swing as follows:
Gain in Time per Swing = (5.00 s / day) / (24 hours / day * 60 minutes / hour * 60 seconds / minute)
Simplifying the units, we get:
Gain in Time per Swing = (5.00 s / 1) / (24 * 60 * 60)
Now, let's calculate:
Gain in Time per Swing = 5.00 / (24 * 60 * 60) s ≈ 5.79 × 10^(-6) s
Since we want to find the fractional change in length, we can use the equation:
Fractional Change in Length = (Gain in Time per Swing) / (Original Period)
To determine the original period, we divide the gain in time per swing by the gain in time per second:
Original Period = (Gain in Time per Swing) / (Gain in Time per Second)
The gain in time per second can be found by multiplying the gain in time per swing by the frequency (swings per second). Since one swing corresponds to half a period, the frequency will be 1/2 times the reciprocal of the period.
Finally, the equation for the fractional change in length becomes:
Fractional Change in Length = (Gain in Time per Swing) / [(Gain in Time per Swing) / (Gain in Time per Second)]
Let's calculate the fractional change in length using the given information:
Gain in Time per Second = (Gain in Time per Swing) * (Frequency)
Frequency = 1 / (2 * Original Period)
Substituting the values, we have:
Gain in Time per Second = (5.79 × 10^(-6) s) * [1 / (2 * Original Period)]
Now, we can calculate the fractional change in length using:
Fractional Change in Length = (Gain in Time per Swing) / (Gain in Time per Second)
Substituting the values, we get:
Fractional Change in Length = (5.79 × 10^(-6) s) / (Gain in Time per Second)
By solving these equations and substituting the appropriate values, the resulting fractional change in pendulum length can be determined.