Question

A heavy wooden ball is hanging from a ceiling by a piece of string that is attached to the ceiling and to the top of the ball. A similar piece of string

Short answer

Answer: The length of the string from the top of the ball to the ceiling is 3 meters.

Step-by-step solution

1. Identify the given information

We are given the following information: - The total length of the string from the ceiling to the bottom of the ball is 4 meters. - The radius of the ball is 0.5 meters.

2. Formulate the problem using the given information

The goal is to find the length of the string from the top of the ball to the ceiling.

3. Calculate the diameter of the ball

We know the radius of the ball is 0.5 meters. To find the diameter, we can use the formula: Diameter = 2 * Radius Diameter = 2 * 0.5 = 1 meter

4. Calculate the length of the string from the top of the ball to the ceiling

Now, we have the diameter of the ball. To find the length of the string between the ball and the ceiling, we can subtract the diameter from the total length of the string: Length of string to ceiling = Total length of string - Diameter of ball Length of string to ceiling = 4 meters - 1 meter = 3 meters Therefore, the length of the string from the top of the ball to the ceiling is 3 meters.

Key concepts

Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic motion or oscillation motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. Imagine a ball attached to a spring; when you pull the ball and let go, it oscillates back and forth. This is a classic example of SHM.

SHM is characterized by its amplitude, frequency, period, and phase. These determine the size of the oscillations, how often they occur, how long one full oscillation takes, and the motion's starting point in its cycle, respectively. Mathematics plays a crucial role in describing SHM through the use of sine and cosine functions, often portrayed as \(x(t) = A\cos(\omega t + \phi)\), where \(x(t)\) is the displacement as a function of time, \(A\) is the amplitude, \(\omega\) is the angular frequency, and \(\phi\) is the phase constant.

SHM is fundamental in physics because it models many natural phenomena, including aspects of pendulum motion. A deep understanding of SHM can greatly enhance a student's grasp of various physical systems and their behaviors.

Pendulum Motion

Pendulum motion is an application of simple harmonic motion and involves the swinging of a weight, or bob, fixed on the end of a string, such that it moves back and forth under the influence of gravity. A simple pendulum is an idealization comprising a weight suspended by a string or rod with negligible mass.

The motion of a pendulum is governed by its physical properties, like the length of the string and the force of gravity acting upon it. The time it takes for the pendulum to complete one full back-and-forth swing is known as its period. A notable point is that the period of a simple pendulum is proportional to the square root of the length of the pendulum string and is given by the formula \(T = 2\pi\sqrt{\frac{l}{g}}\), where \(T\) is the period, \(l\) is the length of the pendulum string, and \(g\) is the acceleration due to gravity.

Pendulum motion is a classic example used to illustrate periodic motion and energy conservation, showcasing how potential energy converts to kinetic energy and back, in a continuous exchange during the oscillation.

Length of Pendulum String

The length of a pendulum string plays a vital role in the characteristics of the pendulum's motion. It's directly related to the period of the pendulum, as mentioned in the discussion on pendulum motion. A common misunderstanding is to consider the length of the pendulum as the distance from the point of suspension to the center of the mass (the bob). However, it is the distance from the point of suspension to the point of force application - which in the case of a simple pendulum is the center of mass of the bob.

In practical problems, like the textbook exercise, determining the correct length of the pendulum string is crucial. The string's length affects how fast or slow the pendulum swings. A longer string results in a slower swing with a longer period, while a shorter string leads to a quicker swing and a shorter period. Therefore, accurate measurement of the length is key to studying pendulum motion and applying knowledge of simple harmonic motion to real-world scenarios.

Improving understanding in these exercises involves clearly distinguishing between the entire length of the pendulum string and the effective length, which does not include the radius or diameter of the swinging bob, as shown in the step-by-step solution.