The number of oscillations made by a pendulum in a given time is inversely proportional to the length of the pendulum. A certain clock with a 20-in.-long pendulum is losing 11.00 min/day. Should the pendulum be lengthened or shortened, and by how much?
To solve this problem, we can set up a proportion to relate the number of oscillations to the length of the pendulum.
Letting x be the number of oscillations in a given time, t be the time taken for these oscillations, and L be the length of the pendulum, we can write the proportion as:
x/L = k/t
where k is a constant of proportionality.
Given that the clock is losing 11.00 minutes per day, we can convert this to minutes per oscillation by dividing 11.00 by the number of oscillations in a day (24 hours * 60 minutes per hour).
Let's calculate the number of oscillations in a day:
Number of oscillations in a day = (24 hours * 60 minutes per hour) / t
We can replace the number of oscillations in a day in the proportion to get:
(24 hours * 60 minutes per hour) / tL = k/t
Simplifying, we have:
1440 / L = k/t
Since k is a constant, we can rewrite this as:
1440 = (k/t) * L
Now, let's consider the current situation with the clock:
Number of oscillations currently = (24 hours * 60 minutes per hour) / (11.00 minutes per day)
L_current = 20 inches
Plugging in the values, the proportion becomes:
(24 * 60) / (11.00) = (k/t) * 20
Simplifying further, we have:
2880 / 11.00 = (k/t) * 20
Now solve for (k/t):
(k/t) = (2880 / 11.00) / 20
(k/t) ≈ 12.545
Substituting this back into the proportion, we have:
1440 = 12.545 * L
Now we can solve for L, the new length of the pendulum:
L = 1440 / 12.545
L ≈ 114.88 inches
Since L_new is greater than L_current, the pendulum should be lengthened. The amount by which it should be lengthened is approximately 114.88 - 20 = 94.88 inches.
To determine whether the pendulum should be lengthened or shortened, we need to understand the relationship between the number of oscillations and the length of the pendulum.
According to the given information, the number of oscillations made by a pendulum in a given time is inversely proportional to the length of the pendulum. In simpler terms, as the length of the pendulum increases, the number of oscillations decreases, and vice versa.
Now, let's analyze the problem:
1. The clock is losing 11.00 minutes per day.
2. The clock has a 20-inch long pendulum.
To find out whether the pendulum should be lengthened or shortened, we need to determine whether the clock is losing time because it's running too slow (fewer oscillations) or too fast (more oscillations).
If the clock is losing time, it means it's running too slow, which suggests that the pendulum is taking longer to complete its oscillations than it should. Therefore, we can conclude that the pendulum needs to be shortened to increase the number of oscillations and compensate for the lost time.
To calculate how much the pendulum should be shortened, we'll use the concept of inverse proportionality:
Let x represent the change in the length of the pendulum (in inches).
Since the length and the number of oscillations are inversely proportional, their product is constant:
Length × Oscillations = constant
Initially, the length of the pendulum is 20 inches, and we can assume the initial number of oscillations is k (a constant) in a given time period.
20 × k = constant
Now, let's consider the new length of the pendulum after shortening by x inches (20 - x):
(20 - x) × Y = constant
Since the clock is losing 11.00 minutes per day, the new number of oscillations in a day (Y) must be (k + 11). We add 11 to compensate for the lost time.
(20 - x) × (k + 11) = constant
Now, we can solve the equation to find the value of x.
(20 - x) × (k + 11) = 20 × k
Expanding the equation:
20k + 220 - xk - 11x = 20k
Simplifying the equation by canceling out similar terms:
220 - xk - 11x = 0
Rearranging the equation:
-xk - 11x = -220
Factoring out -x:
x(k + 11) = 220
Dividing both sides by (k + 11):
x = 220/(k + 11)
To determine the value of x, we need to know the constant k, which corresponds to the initial number of oscillations in a given time period (e.g., a day). Unfortunately, this value is missing in the given information. Therefore, without knowing the value of k, we cannot conclude precisely how much the pendulum should be shortened.
In conclusion, to determine the precise amount by which the pendulum should be shortened, we need the value of k, representing the initial number of oscillations in a day.