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Chapter 6: Problem 70

Figure \(6-53\) shows a conical pendulum, in which the bob (the small object at the lower end of the cord) moves in a horizontal circle at constant speed. (The cord sweeps out a cone as the bob rotates.) The bob has a mass of \(0.040 \mathrm{~kg},\) the string has length \(L=0.90 \mathrm{~m}\) and negligible mass, and the bob follows a circular path of circumference \(0.94 \mathrm{~m} .\) What are (a) the tension in the string and (b) the period of the motion?

Short Answer

Expert verified
Tension in the string is approximately 0.398 N; the period is about 0.927 s.

Step by step solution

01

Find the radius of the circular path

The circumference of a circle is given by the formula \( C = 2\pi r \), where \( r \) is the radius. Rearrange to find the radius: \( r = \frac{C}{2\pi} = \frac{0.94}{2\pi} \approx 0.1497 \) m.
02

Calculate the vertical angle of the string

The string sweeps out a cone, and we need to find the angle \( \theta \) the string makes with the vertical. Use the equation \( \cos \theta = \frac{h}{L} \), where \( h \) is the vertical height of the circle. By Pythagorean theorem in the vertical triangle: \( h = \sqrt{L^2 - r^2} = \sqrt{0.90^2 - 0.1497^2} \approx 0.8852 \) m. Then, \( \cos \theta = \frac{0.8852}{0.90} \approx 0.9836 \).
03

Calculate the tension in the string

The tension \( T \) has vertical and horizontal components balancing weight and providing centripetal force, respectively. Use \( T \cos \theta = mg \) to find \( T \). Rearrange to find \( T = \frac{mg}{ ext{cos} \theta} = \frac{0.040 \times 9.8}{0.9836} \approx 0.398 \) N.
04

Calculate the period of the motion

The centripetal force \( F_c \) is provided by the horizontal component of tension: \( T \sin \theta = \frac{mv^2}{r} \). First, find the speed \( v \) from \( v = \frac{C}{T} \). Rearrange centripetal equation to find speed: \( v = \sqrt{\frac{T \sin \theta \cdot r}{m}} \approx \sqrt{\frac{0.398 \cdot 0.1795 \cdot 0.1497}{0.040}} \approx 1.014 \) m/s. Use \( T = \frac{C}{v} = \frac{0.94}{1.014} \approx 0.927 \) s.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tension in Physics
In the context of a conical pendulum, tension is a crucial concept to understand. Tension refers to the force exerted along the string or rope that connects the pendulum bob to its pivot point. Here, tension acts to keep the bob moving in its circular path while also counteracting gravitational force.

In this particular setup, the tension has two components:
  • Vertical component: Balances the gravitational force acting on the bob, given as \( T \cos \theta = mg \), where \( T \) is the tension in the string, \( \theta \) is the angle with the vertical, \( m \) is the mass of the bob, and \( g \) is the acceleration due to gravity.
  • Horizontal component: Provides the centripetal force required for circular motion, calculated as \( T \sin \theta = \frac{mv^2}{r} \), where \( v \) is the speed of the bob and \( r \) is the radius of the circle.
By solving these equations, one can determine the exact tension in the string, ensuring both components harmoniously allow the pendulum to rotate in its path.
Simple Harmonic Motion
While the motion of a conical pendulum might not be simple harmonic in the traditional sense, it does share some similarities. Simple harmonic motion (SHM) is characterized by oscillation back and forth about an equilibrium position. A conical pendulum oscillates in a circular path but there is a rhythmic repeated motion in nature.

In SHM:
  • There is a restoring force that moves objects back to a central position, proportional to the displacement.
  • Systems such as springs and pendulums exhibit these periodic characteristics.
The circular repetitive nature of a conical pendulum can remind one of SHM, even if the motion isn't purely linear. Observing this, it helps in visualizing how cyclical motions can translate between different types of systems.
Circular Motion
Circular motion is at the heart of a conical pendulum's operation. This type of motion occurs when an object moves along the circumference of a circle at constant speed while continuously changing direction under an inward force.

Key aspects of circular motion include:
  • Constant speed: Although speed remains consistent, the velocity changes due to direction alteration.
  • Inward net force: Essential to maintain circular motion, acting towards the center of the circle.
In the case of a conical pendulum, the circular path is maintained by the horizontal component of tension in the string, ensuring the bob remains on a constant path with unchanged speed. Consistently adjusting direction, due to inward tension forces, is the hallmark of this rotating system.
Centripetal Force
Centripetal force is integral to maintaining objects in circular motion. In a conical pendulum, this force is provided by the horizontal component of the tension in the string.

Understanding centripetal force involves recognizing:
  • It acts perpendicular to the motion, directed towards the center of the circle, maintaining the object's path.
  • For the pendulum, it balances out with the radial component of tension, described by \( T \sin \theta = \frac{mv^2}{r} \).
  • The presence of centripetal force ensures acceleration remains directed inward, changing the bob's direction without affecting its speed.
Recognizing how centripetal force operates helps in understanding why the pendulum maintains its circular path, with net forces harmonizing to continually pull the bob inward as it moves, rather than slowing its speed.

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Most popular questions from this chapter

A bolt is threaded onto one end of a thin horizontal rod, and the rod is then rotated horizontally about its other end. An engineer monitors the motion by flashing a strobe lamp onto the rod and bolt, adjusting the strobe rate until the bolt appears to be in the same eight places during each full rotation of the rod (Fig. \(6-42\) ). The strobe rate is 2000 flashes per second; the bolt has mass \(30 \mathrm{~g}\) and is at radius \(3.5 \mathrm{~cm} .\) What is the magnitude of the force on the bolt from the rod?

A cat dozes on a stationary merry-go-round in an amusement park, at a radius of \(5.4 \mathrm{~m}\) from the center of the ride. Then the operator turns on the ride and brings it up to its proper turning rate of one complete rotation every \(6.0 \mathrm{~s}\). What is the least coefficient of static friction between the cat and the merry-go-round that will allow the cat to stay in place, without sliding (or the cat clinging with its claws)?

A slide-loving pig slides down a certain \(35^{\circ}\) slide in twice the time it would take to slide down a frictionless \(35^{\circ}\) slide. What is the coefficient of kinetic friction between the pig and the slide?

A sling-thrower puts a stone \((0.250 \mathrm{~kg})\) in the sling's pouch \((0.010 \mathrm{~kg})\) and then begins to make the stone and pouch move in a vertical circle of radius \(0.650 \mathrm{~m}\). The cord between the pouch and the person's hand has negligible mass and will break when the tension in the cord is \(33.0 \mathrm{~N}\) or more. Suppose the slingthrower could gradually increase the speed of the stone. (a) Will the breaking occur at the lowest point of the circle or at the highest point? (b) At what speed of the stone will that breaking occur?

In designing circular rides for amusement parks, mechanical engineers must consider how small variations in certain parameters can alter the net force on a passenger. Consider a passenger of mass \(m\) riding around a horizontal circle of radius \(r\) at speed \(v .\) What is the variation \(d F\) in the net force magnitude for (a) a variation \(d r\) in the radius with \(v\) held constant, (b) a variation \(d v\) in the speed with \(r\) held constant, and (c) a variation \(d T\) in the period with \(r\) held constant?
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