Chapter 5: Problem 18
A conical pendulum consists of a bob (mass \(m\) ) attached to a string (length \(L\) ) swinging in a horizontal circle (Fig. 5.11 ). As the string moves, it sweeps out the area of a cone. The angle that the string makes with the vertical is \(\phi\) (a) What is the tension in the string? (b) What is the period of the pendulum?
Short Answer
Expert verified
Tension: \(T = \frac{mg}{\cos(\phi)}\); Period: \(T = 2\pi \sqrt{\frac{L \cos(\phi)}{g}}\).
Step by step solution
01
Understanding the Forces Involved
In a conical pendulum, the tension in the string, gravity, and the centripetal force must be in equilibrium. The tension in the string has two components: one balancing the weight of the bob and the other providing the centripetal force for circular motion.
02
Finding the Vertical Component of Tension
The vertical component of the tension, \( T \), balances the gravitational force. Therefore, we have the equation: \( T \cos(\phi) = mg \).
03
Finding the Horizontal Component of Tension
The horizontal component of the tension provides the centripetal force for the circular motion. Hence: \( T \sin(\phi) = \frac{mv^2}{r} \), where \( r = L \sin(\phi) \).
04
Solving the Tension Equation
We already know that \( T \cos(\phi) = mg \). By rearranging: \( T = \frac{mg}{\cos(\phi)} \). Substituting this into the horizontal force equation, we solve for \( T \).
05
Determining the Velocity
Substitute \( T = \frac{mg}{\cos(\phi)} \) into the horizontal component equation: \[ \frac{mg}{\cos(\phi)} \sin(\phi) = \frac{mv^2}{L \sin(\phi)} \]Canceling \(m\) and rearranging, we find:\[ v^2 = gL \sin^2(\phi) \tan(\phi) \] Thus,\[ v = \sqrt{gL \sin^2(\phi) \tan(\phi)} \]
06
Calculating the Period of the Pendulum
The period \( T (\text{period}) \) can be given by the circumference of the circle divided by the velocity \( v \). The circumference is \( 2\pi L \sin(\phi) \), giving us:\[ T = \frac{2\pi L \sin(\phi)}{v} \]Substituting the expression of \( v \), we get:\[ T = \frac{2\pi L \sin(\phi)}{\sqrt{gL \sin^2(\phi) \tan(\phi)}} \].This simplifies to:\[ T = 2\pi \sqrt{\frac{L \cos(\phi)}{g}} \]
07
Final Expression for Tension and Period
Thus, the tension in the string is given by \( T = \frac{mg}{\cos(\phi)} \) and the period \( T\) of the pendulum is:\[ T = 2\pi \sqrt{\frac{L \cos(\phi)}{g}} \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Tension in the String
In a conical pendulum, the tension in the string is a critical force that keeps the bob in motion. The tension can be thought of as the pull exerted by the string on the bob, and it has two important components:
* **Vertical Component:** This component of tension acts to balance the force of gravity pulling downwards on the bob. Mathematically, this is expressed as \( T \cos(\phi) = mg \). Here, \( T \) is the tension, \( \phi \) is the angle the string makes with the vertical, and \( mg \) is the gravitational force on the bob.
* **Horizontal Component:** This component provides the centripetal force necessary for the bob's circular motion. It is given by \( T \sin(\phi) = \frac{mv^2}{r} \), where \( r \) is the radius of the circular path the bob follows, calculated as \( r = L \sin(\phi) \).
By understanding and calculating these components, one can find the total tension, which is accomplished by solving the equation \( T = \frac{mg}{\cos(\phi)} \). This shows how tension spans both gravitational balance and the requirements of circular motion.
* **Vertical Component:** This component of tension acts to balance the force of gravity pulling downwards on the bob. Mathematically, this is expressed as \( T \cos(\phi) = mg \). Here, \( T \) is the tension, \( \phi \) is the angle the string makes with the vertical, and \( mg \) is the gravitational force on the bob.
* **Horizontal Component:** This component provides the centripetal force necessary for the bob's circular motion. It is given by \( T \sin(\phi) = \frac{mv^2}{r} \), where \( r \) is the radius of the circular path the bob follows, calculated as \( r = L \sin(\phi) \).
By understanding and calculating these components, one can find the total tension, which is accomplished by solving the equation \( T = \frac{mg}{\cos(\phi)} \). This shows how tension spans both gravitational balance and the requirements of circular motion.
Centripetal Force
Centripetal force is the central force that keeps the bob of a conical pendulum moving in its constant circular path. Rather than being a separate force on its own, it results from the horizontal component of tension in this context. To understand the centripetal force in action:
* **Defined by Tension:** It arises because the tension in the string has a component acting towards the center of the circle, ensuring the bob remains on its circular trajectory.
* **Expressed Mathematically:** The centripetal force provided by the string can be expressed as \( T \sin(\phi) = \frac{mv^2}{r} \). The term \( \frac{mv^2}{r} \) is a classic expression for centripetal force, indicating the dependence on mass \( m \), velocity \( v \), and radius \( r \).
In simpler terms, the centripetal force is essential for the bob's motion and dictates the relationship between the pendulum's tension, the circular path's requirements, and the bob's speed.
* **Defined by Tension:** It arises because the tension in the string has a component acting towards the center of the circle, ensuring the bob remains on its circular trajectory.
* **Expressed Mathematically:** The centripetal force provided by the string can be expressed as \( T \sin(\phi) = \frac{mv^2}{r} \). The term \( \frac{mv^2}{r} \) is a classic expression for centripetal force, indicating the dependence on mass \( m \), velocity \( v \), and radius \( r \).
In simpler terms, the centripetal force is essential for the bob's motion and dictates the relationship between the pendulum's tension, the circular path's requirements, and the bob's speed.
Period of Pendulum
The period of a pendulum refers to the time it takes to complete one full swing, or oscillation, around the circle. For a conical pendulum, some key points to remember about its period include:
* **Calculation Method:** The period \( T \) is determined by dividing the circumference of the circle by the bob's velocity. Mathematically, this is \( T = \frac{2\pi L \sin(\phi)}{v} \). The period can be further simplified using the expression we derived for velocity, resulting in \( T = 2\pi \sqrt{\frac{L \cos(\phi)}{g}} \).
* **Independence from Mass:** Interestingly, the period is independent of the mass of the bob. It mainly depends on the length of the string \( L \), the angle \( \phi \), and the gravitational constant \( g \).
This means, whether the bob is heavy or light, the time taken for one complete cycle remains the same as long as the angle and string length remain unchanged.
* **Calculation Method:** The period \( T \) is determined by dividing the circumference of the circle by the bob's velocity. Mathematically, this is \( T = \frac{2\pi L \sin(\phi)}{v} \). The period can be further simplified using the expression we derived for velocity, resulting in \( T = 2\pi \sqrt{\frac{L \cos(\phi)}{g}} \).
* **Independence from Mass:** Interestingly, the period is independent of the mass of the bob. It mainly depends on the length of the string \( L \), the angle \( \phi \), and the gravitational constant \( g \).
This means, whether the bob is heavy or light, the time taken for one complete cycle remains the same as long as the angle and string length remain unchanged.
Gravitational Force
Gravitational force is the force of attraction between objects with mass. In a conical pendulum system, gravity plays a fundamental role:
* **Downward Pull:** Gravity acts vertically downward on the pendulum bob with a force equal to its weight, or \( mg \), where \( m \) is the mass of the bob, and \( g \) is the acceleration due to gravity.
* **Counterbalanced by Tension:** The vertical component of the string's tension balances this gravitational force so that the bob remains in horizontal motion and doesn't just drop straight down. This relationship is given by \( T \cos\phi = mg \).
The gravitational force is crucial in determining both the equilibrium of forces and the tension required in the string to maintain the pendulum's circular motion.
* **Downward Pull:** Gravity acts vertically downward on the pendulum bob with a force equal to its weight, or \( mg \), where \( m \) is the mass of the bob, and \( g \) is the acceleration due to gravity.
* **Counterbalanced by Tension:** The vertical component of the string's tension balances this gravitational force so that the bob remains in horizontal motion and doesn't just drop straight down. This relationship is given by \( T \cos\phi = mg \).
The gravitational force is crucial in determining both the equilibrium of forces and the tension required in the string to maintain the pendulum's circular motion.
Circular Motion
The conical pendulum exemplifies circular motion, where an object moves in a circular path due to internal and balancing forces. Here's how it works in this system:
* **Constant Radius:** The bob travels in a circle with a radius \( r = L \sin(\phi) \), a constant as long as \( \phi \) and \( L \) remain unaltered.
* **Uniform Motion:** Although the direction is continuously changing, speed remains constant, creating a uniform circular motion.
* **Force Balance:** For the bob to maintain this motion, the horizontal component of tension must supply the necessary centripetal force. This ensures that any force acting outward due to circular motion is countered, keeping the bob in its path.
Circular motion in a conical pendulum is a beautiful demonstration of physics in action, showing how forces cohesively maintain a steady path.
* **Constant Radius:** The bob travels in a circle with a radius \( r = L \sin(\phi) \), a constant as long as \( \phi \) and \( L \) remain unaltered.
* **Uniform Motion:** Although the direction is continuously changing, speed remains constant, creating a uniform circular motion.
* **Force Balance:** For the bob to maintain this motion, the horizontal component of tension must supply the necessary centripetal force. This ensures that any force acting outward due to circular motion is countered, keeping the bob in its path.
Circular motion in a conical pendulum is a beautiful demonstration of physics in action, showing how forces cohesively maintain a steady path.
