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Chapter 8: Problem 24

(I) A 52-kg person riding a bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 cm. \((a)\) What is the maximum torque she exerts? \((b)\) How could she exert more torque?

Short Answer

Expert verified
(a) Maximum torque is 86.632 Nm. (b) More torque can be exerted by increasing the force applied or the pedal radius.

Step by step solution

01

Understand Torque Concept

Torque is the measure of the rotational force applied to an object. It is calculated by the product of force applied and the distance from the pivot point (radius). The formula for torque is \( \tau = F \times r \times \sin(\theta) \), where \( F \) is the force, \( r \) is the distance from the pivot point, and \( \theta \) is the angle between force direction and lever arm.
02

Identify the Given Values

The problem provides the following values: weight of the person \( m = 52 \) kg, radius of the pedal \( r = 17 \) cm (which is 0.17 m for calculation consistency in SI units), and acceleration due to gravity \( g = 9.8 \) m/s\(^2\). Since weight is applied directly downward, \( \theta = 90^{\circ} \), and \( \sin(90^{\circ}) = 1 \).
03

Calculate the Force Exerted

The force exerted by the person is her weight, calculated by \( F = m \times g = 52 \times 9.8 = 509.6 \) N.
04

Calculate Maximum Torque

Substitute the calculated force \( F = 509.6 \) N and the radius \( r = 0.17 \) m into the torque formula: \( \tau = 509.6 \times 0.17 \times 1 = 86.632 \) Nm. Therefore, the maximum torque exerted is 86.632 Nm.
05

Consider Increasing the Torque

The torque can be increased in two main ways: (1) increase the force applied, either by increasing weight or through additional force, or (2) increase the distance from the pivot by using a pedal with a longer radius arm.

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Key Concepts

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

Rotational Force
Rotational force, often called torque, is an important concept in physics, especially when studying how objects turn. It is simply the ability of a force to cause an object to rotate around an axis. This idea is crucial for understanding mechanics and motion.

Think of torque as a twisting force. The amount of rotation depends on:
  • The force applied
  • The distance from the pivot or fulcrum (where the object rotates)
  • The angle at which the force is applied
The standard formula for torque is \( \tau = F \times r \times \sin(\theta) \), where:
  • \( \tau \) is the torque or rotational force
  • \( F \) is the force applied
  • \( r \) is the radius or the distance from the center of rotation
  • \( \theta \) is the angle between the force and the lever arm
Overall, understanding rotational force helps in solving many physics problems regarding movement and mechanical advantage.
Physics Problems
In physics, problems involving torque can be solved by first breaking down the components involved. Understanding how to approach these problems requires recognizing the forces acting on the system and how they interact to cause rotation.

When faced with a physics problem about torque, follow these steps:
  • Identify all forces acting on the object
  • Determine the line of action of these forces
  • Measure the distance from the axis of rotation to where the force is applied
  • Calculate the angle of the applied force if it is not perpendicular
  • Use the torque formula to evaluate the rotational effect
Having a clear understanding of physics problems involving torque can simplify the process of finding solutions and help apply the concept to real-world scenarios.
Torque Calculation
Calculating torque involves using the formula \( \tau = F \times r \times \sin(\theta) \). This example will help clarify the process.

Let's consider a 52-kg person who is cycling and applying their full weight to the pedal. The parameters given are:
  • Weight of the person: 52 kg, which translates to a force of \( F = m \times g = 52 \times 9.8 = 509.6 \) N
  • Radius of the pedal's circular path: 0.17 m
  • The angle \( \theta \) is 90 degrees because the force is applied perpendicularly
Using the torque formula, the calculation becomes:\[ \tau = 509.6 \times 0.17 \times \sin(90^{\circ}) = 86.632 \, \text{Nm} \]This calculation shows that the maximum torque exerted by the cyclist is 86.632 Nm. Calculating torque precisely helps in optimizing mechanical systems and understanding their efficiency.
Pedal Mechanics
Pedal mechanics is essential for understanding how bicycles utilize torque to move forward. When a cyclist pedals, they apply force that results in rotation of the crank arms, translating to the wheel's motion.

Key aspects of pedal mechanics include:
  • Application of force: The force applied onto the pedals is critical for generating maximum torque.
  • Length of crank arm: A longer crank arm increases the radius \( r \), thus enhancing torque without requiring more force.
  • Pedaling angle: The angle \( \theta \) impacts how effectively the force contributes to torque. Perpendicular force maximizes this effect.
By adjusting these elements, a cyclist can maximize their efficiency, especially when climbing hills or accelerating. Understanding pedal mechanics empowers cyclists to improve their performance through informed technique and equipment choices.

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Most popular questions from this chapter

(II) Estimate the kinetic energy of the Earth with respect to the Sun as the sum of two terms, \((a)\) that due to its daily rotation about its axis, and \((b)\) that due to its yearly revolution about the Sun. [Assume the Earth is a uniform sphere with \(mass = 6.0 \times 10^{24} kg\), \(radius = 6.4 \times 10^6 m\), and is \(1.5 \times 10^8 km\) from the Sun.]

(I) An automobile engine slows down from 3500 rpm to 1200 rpm in 2.5 s. Calculate \((a)\) its angular acceleration, assumed constant, and \((b)\) the total number of revolutions the engine makes in this time.

(II) A small rubber wheel is used to drive a large pottery wheel. The two wheels are mounted so that their circular edges touch. The small wheel has a radius of 2.0 cm and accelerates at the rate of \(7.2 rad/s^2\), and it is in contact with the pottery wheel (radius 27.0 cm) without slipping. Calculate \((a)\) the angular acceleration of the pottery wheel, and \((b)\) the time it takes the pottery wheel to reach its required speed of 65 rpm.

(II) A person exerts a horizontal force of 42 N on the end of a door 96 cm wide. What is the magnitude of the torque if the force is exerted \((a)\) perpendicular to the door and \((b)\) at a 60.0\(^{\circ}\) angle to the face of the door?

\((a)\) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.013 m. Use conservation of energy to calculate the linear speed of the yo-yo just before it reaches the end of its 1.0-m-long string, if it is released from rest. \((b)\) What fraction of its kinetic energy is rotational?
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