Problem 69 - University Physics with Modern Physics

Question

A grandfather clock uses a pendulum and a weight. The pendulum has a period of \(2.00 \mathrm{~s}\), and the mass of the bob is 250. \(\mathrm{g}\). The weight slowly falls, providing the energy to overcome the damping of the pendulum due to friction. The weight has a mass of \(1.00 \mathrm{~kg}\), and it moves down \(25.0 \mathrm{~cm}\) every day. Find \(Q\) for this clock. Assume that the amplitude of the oscillation of the pendulum is \(10.0^{\circ}\)

Step-by-Step Solution

Answer

Answer: The quality factor (Q) for the given grandfather clock is approximately 3760.

Step 1: Calculate the initial total energy of the pendulum

We have the mass of the bob (m_bob = 250 g, which is 0.25 kg), and the amplitude of oscillation (10°). We'll first convert the angle to radians: 10° × (π/180) = 10π/180 ≈ 0.174 radians We need to find the length of the pendulum (l) using the period formula T = 2π √(l/g), where g is the gravitational acceleration (9.81 m/s²). We will rearrange the formula to calculate l: l = (T^2 * g) / (4π^2) Given T = 2 seconds, the length of the pendulum can be calculated as: l ≈ ((2^2 * 9.81) / (4π^2)) ≈ 1.005 m The initial total energy E can be calculated as the gravitational potential energy E = m_bob * g * h, where h = l * (1 - cos(amplitude)). E ≈ 0.25 * 9.81 * 1.005 * (1 - cos(0.174)) ≈ 0.034 J (Joules)

Step 2: Calculate the energy lost due to friction each day

The weight of the clock moves downward 25 cm (0.25 m) every day. We can find the gravitational potential energy lost by the weight each day as E(f) = m_weight * g * h_movement, where m_weight = 1.00 kg and h_movement = 0.25 m. E(f) ≈ 1 * 9.81 * 0.25 ≈ 2.453 J (Joules)

Step 3: Calculate the energy lost per cycle

To find the energy lost during each cycle, we need to know how many cycles the pendulum completes in a day. There are 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute, so the total seconds in a day are: Total_seconds = 24 * 60 * 60 = 86,400 s The period of the pendulum is 2.00 s, so the number of cycles completed in a day is: No_of_cycles = Total_seconds / T = 86,400 / 2 = 43,200 cycles Thus, the energy lost per cycle e_lost can be calculated as: e_lost = E(f) / No_of_cycles = 2.453 J / 43,200 ≈ 5.676 × 10^-5 J

Step 4: Calculate the quality factor (Q) for the clock

Now that we have the initial total energy (E) and the energy lost per cycle (e_lost), we can find the quality factor Q using the formula: Q = 2π * (E / e_lost) Q ≈ 2π * (0.034 J / 5.676 × 10^-5 J) ≈ 3760 The quality factor (Q) for this grandfather clock is approximately 3760.

Key Concepts

Harmonic motion

Harmonic motion is a type of periodic motion where the restoring force is proportional to the displacement. It's like the swinging motion of a pendulum. In this context, pendulums are excellent examples, as they naturally exhibit harmonic motion. When the pendulum is released from a small angle, it swings back and forth.
The motion is predictable and described by simple harmonic motion (SHM) equations. Since it relies on gravity, the period (or time it takes for one full swing back and forth) of a simple pendulum is given by the formula:

\(T = 2\pi \sqrt{\frac{l}{g}}\)

• Where \(T\) is the period, \(l\) is the length of the pendulum, and \(g\) is the acceleration due to gravity.

One crucial aspect of harmonic motion is its dependency on factors like length, meaning a longer pendulum takes more time to complete a cycle. Harmonic motion is key to understanding natural rhythms in physics and engineering, providing insights into how energy travels through various systems.

Energy dissipation

Energy dissipation refers to the loss of energy in a system, typically due to friction or air resistance, which can decrease the amplitude of motion over time. In a pendulum, energy dissipation gradually slows down its swing.
Friction in the pivot and air resistance cause this energy loss.
Imagine a pendulum in a clock. It needs continuous input of energy to maintain its regular motion.

• The weight attached to the clock descends day by day, compensating for the energy lost to friction.
• The energy dissipated each day is equivalent to the gravitational potential energy the weight loses as it drops.

Without compensation for energy dissipation, the pendulum would stop swinging. This process highlights how real-world systems can't operate indefinitely without external energy sources. Understanding energy dissipation helps in designing systems that optimize energy usage, mitigating unnecessary losses.

Periodic motion

Periodic motion is movement that repeats at regular time intervals. It's at the heart of many natural and mechanical processes, such as the ticking of a clock or the oscillation of a pendulum.
For a pendulum, its periodic motion is what keeps time, completing consistent swings of equal duration.

• In our exercise, the pendulum's period is 2 seconds, meaning it swings forth and back consistently in this timeframe.
• This regularity allows for its application in timekeeping devices like grandfather clocks.

The beauty of periodic motion lies in its predictability and stability, which is why it's so valuable in science and technology. Knowing how periodic systems assume a steady rhythm allows us to measure and utilize these patterns in everyday life, from electricity to musical instruments. Periodic motion serves as a fundamental principle for constructing mechanisms that require synchronized activities.