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Chapter 12: Problem 29

Which of the following best describes the relationship between the magnitude of the tension force, \(F_{\mathrm{T}},\) in the string of a pendulum and the radial component of gravity that pulls antiparallel to the tension, \(F_{\mathrm{g}}\) radial? Assume that the pendulum is only displaced by a small amount. (A) \(F_{\mathrm{T}}>F_{\mathrm{g}, \text { radial }}\) (B) \(F_{\mathrm{T}} \geq F_{\text { g, radial }}\) (C) \(F_{\mathrm{T}}=F_{\mathrm{g}, \text { radial }}\) (D) \(F_{\mathrm{T}} \leq F_{\mathrm{g}, \text { radial }}\)

Short Answer

Expert verified
The correct answer is (C) \(F_{\mathrm{T}}=F_{\mathrm{g, radial}}\) as these forces are in equilibrium at the maximum displacement of the pendulum.

Step by step solution

01

Understand the Forces acting on the Pendulum

A pendulum displaced by a small angle will have two main forces at play. One is \(F_{\mathrm{T}}\), the tension force in the string, which pulls the pendulum bob towards the pivot point. The other is \(F_{\mathrm{g, radial}}\), the component of gravity which is parallel and opposite to \(F_{\mathrm{T}}\). If the displacement is very small, the radial component of gravitational force provides the restoring force to move the pendulum back to its equilibrium position.
02

Compare the Magnitudes of the Forces

When a pendulum is at its maximum displacement (still a small angle), the tension force and the radial component of the gravitational force are in equilibrium. So, 鈬媆(F_{\mathrm{T}}\) equals \(F_{\mathrm{g, radial}}\), not greater or less than.
03

Choose the Correct Option

Therefore, the best description of the relationship of these forces should reflect this balance. As the pendulum swings back towards its equilibrium position, the tension force will become greater than the radial component of gravity. However, the question specifies the situation at the maximum displacement, at which point these forces are balanced. So the correct answer is (C) \(F_{\mathrm{T}}=F_{\mathrm{g, radial}}\).

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Key Concepts

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

Tension Force
The tension force, denoted as \( F_{\mathrm{T}} \), is a fundamental aspect of pendulum motion. It represents the pulling force exerted by a string on the pendulum bob. This force always acts along the string, pulling the bob towards the pivot point.
During the swing of a pendulum, the tension force varies with the pendulum's displacement. When the pendulum is at its highest point, the tension is at its minimum, counteracting gravity's force. At this position, the tension helps to balance the forces acting on the pendulum.
  • *Direction*: Always directed towards the pivot point.
  • *Magnitude*: Changes depending on the pendulum's position and angular displacement.
Radial Component of Gravity
Gravity exerts a constant force on the pendulum. However, it's the radial component of gravity, \( F_{\mathrm{g, radial}} \), that influences the pendulum's swing back to its equilibrium position. This component acts along the direction of the string, opposing the tension.
When the pendulum is displaced, gravity can be broken into components: radial (along the string) and tangential (perpendicular to the string). The radial component keeps the pendulum's path circular and ensures it follows through the arc.
  • *Balancing Act*: At the maximum displacement angle, the radial force is balanced by the tension force.
  • *Role in Restoring*: Always pulling inward, facilitating the swinging motion back.
Equilibrium Position
The equilibrium position of a pendulum is the location where no net force acts perpendicular to the pendulum鈥檚 path. It's the lowest point in the pendulum's swing, where potential energy is minimized, and kinetic energy is maximized. At this point, the pendulum doesn't experience any restorative force as it's not displaced from this central position.
As the pendulum swings through equilibrium, tension and radial gravity momentarily balance. However, due to its velocity, the pendulum overshoots this point, continuing its motion under inertia's influence.
  • *Steady State*: No net torque here; all forces balance out.
  • *Energy Conversion*: Conversion between potential and kinetic energy is most evident.
Small Angle Approximation
The small angle approximation is a simplifying assumption often used in pendulum physics. When the displacement angle \( \theta \) is small (typically less than about 15 degrees), \( \sin(\theta) \) is nearly equal to \( \theta \) (in radians). This approximation allows us to linearize the pendulum's equations of motion.
With this approximation, the pendulum's motion resembles that of simple harmonic oscillators. It brings simplicity to complex trigonometric calculations and is vital in most elementary pendulum problems.
  • *Applicability*: Valid for angles where deviations from linearity are negligible.
  • *Benefits*: Makes complex problems more accessible through linearization.

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Most popular questions from this chapter

If a roller coaster cart of mass \(m\) was not attached to the track, it would still remain in contact with a track throughout a loop of radius \(r\) as long as (A) \(v \leq \sqrt{(r g)}\) (B) \(v \geq \sqrt{(r g)}\) (C) \(v \leq \sqrt{(r g / m)}\) (D) \(v \geq \sqrt{(r g / m)}\)

Given that the Earth's mass is \(m,\) its tangential speed as it revolves around the Sun is \(v\) , and the distance from the Sun to the Earth is \(r,\) which of the following correctly describes the work done by the centripetal force, \(W_{c}\) , in one year's time? (A) \(\mathrm{W}_{\mathrm{c}}>2 r\left(m v^{2} / r\right)\) (B) \(\mathrm{W}_{\mathrm{c}}=2 r\left(m v^{2} / r\right)\) (C) \(\mathrm{W}_{\mathrm{c}}<2 r\left(\mathrm{mv}^{2} / r\right)\) (D) Cannot be determined

A toy car and a toy truck collide. If the toy truck鈥檚 mass is double the toy car鈥檚 mass, then, compared to the acceleration of the truck, the acceleration of the car during the collision will be (A) double the magnitude and in the same direction (B) double the magnitude and in the opposite direction (C) half the magnitude and in the same direction (D) half the magnitude and in the opposite direction

A ball is thrown in a projectile motion trajectory with an initial velocity \(v\) at an angle \(\theta\) above the ground. If the acceleration due to gravity is \(-g,\) which of the following is the correct expression of the time it takes for the ball to reach its highest point, \(y,\) from the ground? (A) \(v^{2} \sin t / \mathrm{g}\) (B) \(-v \cos \theta / g\) (C) \(v \sin \theta / g\) (D) \(v^{2} \cos \theta / g\)

Pretend someone actually managed to dig a hole straight through the center of the Earth all the way to the other side. If an object were dropped down that hole, which of the following would best describe its motion? Assume ideal conditions and that the object cannot be destroyed. (A) It would fall to the center of the Earth and stop there. (B) It would fall through the hole to the other side, continue past the opposite side鈥檚 opening, and fly into space. (C) It would oscillate back and forth from one opening to the other indefinitely. (D) It would fall to the other side and stop there.
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