Problem 47 - College Physics

Question

(a) If a pendulum has period \(T\) and you double its length, what is its new period in terms of \(T ?\) (b) If a pendulum has a length \(L\) and you want to triple its frequency, what should be its length in terms of \(L ?\) (c) Suppose a pendulum has a length \(L\) and period \(T\) on earth. If you take it to a planet where the acceleration of freely falling objects is 10 times what it is on earth, what should you do to the length to keep the period the same as on earth? (d) If you do not change the pendulum's length in part (c), what is its period on that planet in terms of \(T ?\) (e) If a pendulum has a period \(T\) and you triple the mass of its bob, what happens to the period (in terms of \(T\) )?

Step-by-Step Solution

Answer

(a) \(\sqrt{2}T\); (b) \(L/9\); (c) Increase to \(10L\); (d) \(\frac{1}{\sqrt{10}}T\); (e) No change.

Step 1: Understanding the Period of a Pendulum

The period 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, and \(g\) is the acceleration due to gravity.

Step 2: (a) Doubling the Length

When the length of the pendulum is doubled, the new length becomes \(2L\). The new period \(T'\) is:\[ T' = 2\pi \sqrt{\frac{2L}{g}} = 2\pi \sqrt{2} \sqrt{\frac{L}{g}} = \sqrt{2}T \]So, the new period is \(\sqrt{2}T\).

Step 3: (b) Tripling the Frequency

The frequency \(f\) of a pendulum is the reciprocal of the period: \(f = 1/T\). To triple the frequency, we have:\[ 3f = \frac{3}{T} \]Substituting the period formula:\[ 3 \cdot \frac{1}{2\pi \sqrt{\frac{L}{g}}} = \frac{1}{T'} \]Setting the new period \(T' = \frac{T}{3}\), find new length:\[ \frac{T}{3} = 2\pi \sqrt{\frac{L'}{g}} \Rightarrow \frac{T}{3} = T \sqrt{\frac{L'}{L}} \Rightarrow \frac{1}{3} = \sqrt{\frac{L'}{L}} \Rightarrow L' = \frac{L}{9} \]Thus, the new length should be \(L/9\).

Step 4: (c) Adjusting for Different Gravity

On a planet with gravitational acceleration of \(10g\), to maintain the same period \(T\), the new length \(L'\) required is calculated by:\[ T = 2\pi \sqrt{\frac{L}{g}} = 2\pi \sqrt{\frac{L'}{10g}} \]Solving for \(L'\):\[ \sqrt{L} = \sqrt{\frac{L'}{10}} \Rightarrow L = \frac{L'}{10} \Rightarrow L' = 10L \]The length should be increased to \(10L\).

Step 5: (d) Period Without Length Change on the Planet

If no change is made to the length, the new period \(T'\) on the planet with \(10g\) is:\[ T' = 2\pi \sqrt{\frac{L}{10g}} = \frac{1}{\sqrt{10}}T \]The period becomes \(\frac{1}{\sqrt{10}}T\).

Step 6: (e) Effect of Tripling the Mass

The period of a pendulum does not depend on the mass of the bob. Therefore, even if the mass of the bob is tripled, the period remains unchanged: \(T\).

Key Concepts

Simple Pendulum

A simple pendulum consists of a weight, known as the bob, attached to a string or rod of fixed length that swings freely under the influence of gravity. The motion of a simple pendulum exhibits regular, repetitive movement, appearing to swing back and forth with a constant rhythm when displaced from its resting position. The motion depends primarily on two factors: the length of the pendulum and the gravitational force.

One fascinating aspect of simple pendulums is that their period, the time it takes to complete one full oscillation, is independent of the mass of the bob. This means that whether the bob is light or heavy, the time it takes to swing back and forth remains the same, assuming other factors like pendulum length remain unchanged. This principle of uniform oscillation stems from the physics of harmonic motion, which governs how pendulums and even swings work in the real world.

Gravitational Acceleration

Gravitational acceleration, often denoted by the symbol \(g\), plays a crucial role in determining the behavior of a pendulum. It represents the rate at which an object accelerates towards the Earth due to gravity, approximately \(9.81 \, \text{m/s}^2\) on the surface of the Earth. This value, however, can differ if the pendulum is on another planet or at a different altitude, which impacts its period.

In the formula for a pendulum's period, \(T = 2\pi \sqrt{\frac{L}{g}}\), \(g\) directly affects how quickly the pendulum swings. A higher gravitational acceleration results in a shorter period, indicating faster swings, while lower gravitational acceleration prolongs the period, causing slower swings.

• On Earth, this value is constant, but on a planet with higher \(g\), like in our textbook problem, the length of the pendulum needs to be adjusted to maintain a constant period.
• Understanding \(g\)'s influence helps predict how pendulums behave in different gravitational environments.

Pendulum Frequency

The frequency of a pendulum refers to how many complete cycles it makes in one second. It is the inverse of the period, calculating as \(f = \frac{1}{T}\), where \(T\) is the period.

When we talk about designing pendulums, adjustments to their frequency are often desired for specific applications. For example, if you want a pendulum to oscillate more frequently, you can increase the frequency. In practical terms, when you want to triple the pendulum's frequency, you need to shorten the pendulum's length significantly.

Using the period formula \(T = 2\pi \sqrt{\frac{L}{g}}\), we find that any increase in frequency requires a proportional decrease in length: to triple the frequency, the length should be reduced to one-ninth of its original length, allowing it to swing faster. The relations between period, frequency, and length are interconnected, showcasing the precise harmony in pendular motion.

Pendulum Length Adjustment

Length adjustment is a critical aspect when it comes to controlling the characteristics of a simple pendulum. The length directly influences both the period and frequency of oscillation. If you aim to change a pendulum's period or frequency, adjusting its length is an effective way to achieve that without affecting other properties like mass.

Changes in length can have dramatic effects: doubling the length, for instance, increases the period by a factor of \(\sqrt{2}\), and reducing the length to make the pendulum faster changes its frequency substantially. Specific scenarios, such as moving a pendulum to a location with different gravitational pull, require length adjustments to maintain the same period.

• For example, increasing the length on a high-gravity planet compensates for the stronger gravitational acceleration, keeping the oscillation period the same as on Earth.
• Each adjustment is about finding the right balance, factoring in the pendulum's environment and desired motion properties.

Length changes are a simple but powerful way to fine-tune the pendulum's behavior, making it versatile for different experimental and practical purposes.