Suppose that a grandfather clock (a simple pendulum) is running slowly. That is, the time it takes to complete each cycle is longer than it should be. Should one shorten or lengthen the pendulum to make the clock keep the correct time? Why?

To make the grandfather clock keep the correct time, the pendulum should be shortened. Shortening the pendulum decreases its length, which in turn decreases the period of the pendulum.

The period of a pendulum (the time it takes to complete one cycle) is determined by its length. According to the formula 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. As the length of the pendulum decreases, the period also decreases.

Since the clock is running slowly (the time it takes for each cycle is longer), it means that the pendulum is taking more time to swing back and forth. By shortening the pendulum, the time it takes for each swing decreases, which would make the clock keep the correct time.

In summary, shortening the pendulum of the grandfather clock would make it keep the correct time by reducing the period of the pendulum.

To understand whether one should shorten or lengthen the pendulum to make the grandfather clock keep the correct time, let's consider the principle behind the operation of a simple pendulum.

The time it takes for a simple pendulum to complete one full swing, also known as the period, is determined by the length of the pendulum and the acceleration due to gravity. The period of a pendulum can be calculated using the formula:

T = 2π * √(L/g),

where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.

Now, going back to the scenario where the clock is running slowly, there could be several factors causing this issue. If we assume no external factors are involved, such as air resistance, the most likely cause is an incorrect length of the pendulum.

If the clock is running slowly, the period of the pendulum is longer than it should be. According to the formula, if we want to decrease the period, we need to decrease the length of the pendulum (L). Therefore, in this case, one should shorten the pendulum to make the clock keep the correct time.

By shortening the pendulum, the period of the pendulum will decrease, and the swings will occur more quickly, allowing the clock to keep accurate time.

However, it is important to note that this is a simplified explanation, and there could be other factors at play in a real-world scenario. Factors such as air resistance, friction in the clock's mechanism, or irregularities in the swing can also affect the accuracy of the clock.

The period of a pendulum is proportional to the square root of the length of the pendulum. You want it to "tick" (swing back and forth) faster. What does that tell you?

Lengthening the pendulum will increase the frequency and make it run faster