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Chapter 10: Problem 24

A 62 -kg person riding a bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius \(17 \mathrm{~cm} .\) ( \(a\) ) What is the maximum torque she exerts? (b) How could she exert more torque?

Short Answer

Expert verified
The maximum torque is approximately 103.3 Nm. She could exert more torque by using pedals with a larger radius or increasing the force applied.

Step by step solution

01

Understanding the Problem

We need to find the maximum torque exerted when the person applies her full weight on the pedal. Torque is the rotational equivalent of force, and it depends on the force applied and the distance from the pivot point, which here is the axis around which the pedals rotate.
02

Calculate the Force Exerted

The force exerted by the person is equal to her weight since she applies all her weight on each pedal. The weight is calculated using the formula: \[ F = m \cdot g \] where \( m = 62 \text{ kg} \) is the mass and \( g = 9.8 \text{ m/s}^2 \) is the acceleration due to gravity. Thus, the force is \[ F = 62 \cdot 9.8 = 607.6 \text{ N} \]
03

Determine the Radius

The radius of the circle in which the pedals rotate is given as \( 17 \text{ cm} \). Convert this into meters since SI units are required, \[ r = 17 \text{ cm} = 0.17 \text{ m} \]
04

Compute the Maximum Torque

The maximum torque is given by the product of the force and the effective radius at which the force is applied:\[ \tau = F \cdot r \]Substitute the values to calculate the torque:\[ \tau = 607.6 \cdot 0.17 = 103.292 \text{ Nm} \]Thus, the maximum torque she exerts is approximately \( 103.3 \text{ Nm} \).
05

Reasons to Exert More Torque

The torque can be increased by either increasing the force applied or increasing the radius at which the force is applied. In practice, she could exert more torque by either increasing her body weight (increased force) or using pedals with a larger radius.

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Key Concepts

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

Rotational Motion
Rotational motion revolves around objects moving in a circular path around a fixed point or axis. This is different from linear motion, where objects move along a straight line.
In the context of a bicycle, the pedals and cranks are perfect examples of rotational motion in action. As the cyclist applies their weight to the pedal, the pedal goes around the circle attached to the crank axis. This circular path allows pedals to rotate, transferring energy to the bicycle's wheels, eventually causing the bike to move forward.
A key aspect of rotational motion is torque, which is essentially the 'strength' of this rotational force. Torque is influenced by two main factors: the amount of force exerted and the radius at which the force is applied. In this way, rotational motion is not just about moving, but moving with leverage for maximum efficiency.
Force
Force is a fundamental concept in physics and describes any push or pull on an object. In this context, the force is provided by the cyclist's weight being exerted on the pedal.
The force due to gravity on the cyclist's body is commonly referred to as weight, and it can be calculated with the formula:
  • \( F = m \cdot g \)
where \( m \) represents mass and \( g \) is the acceleration due to gravity, approximately \( 9.8 \, \text{m/s}^2 \).
In the given exercise, the cyclist's full body weight is applied as force on each pedal to climb a hill. This is calculated as:
  • \( F = 62 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 607.6 \, \text{N} \)
A higher force on the pedal leads to greater torque, facilitating smoother hill climbing.
Radius
The radius plays an essential role in determining the torque. It is the distance from the center of rotation to the point where the force is applied. In bicycle mechanics, this is the distance from the crank's axis to the end of the pedal.
In the problem presented, the radius is given as 17 cm. However, for calculations, it is crucial to convert it into meters, resulting in \( 0.17 \, \text{m} \). Using the formula:
  • \( \tau = F \cdot r \)
where \( \tau \) is torque, \( F \) is force and \( r \) is the radius.
An increase in radius proportionally increases torque. Consequently, increasing the pedal's radius means that even with the same force applied, more torque can be generated. Conversely, a smaller radius requires more force to achieve the same torque level. Hence, the strategic adjustment of the radius can either minimize effort or maximize power during cycling.

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Most popular questions from this chapter

A softball player swings a bat, accelerating it from rest to \(2.7 \mathrm{rev} / \mathrm{s}\) in a time of \(0.20 \mathrm{~s}\). Approximate the bat as a \(2.2-\mathrm{kg}\) uniform rod of length \(0.95 \mathrm{~m},\) and compute the torque the player applies to one end of it.

(I) Use the parallel-axis theorem to show that the moment of inertia of a thin rod about an axis perpendicular to the rod at one end is \(I=\frac{1}{3} M \ell^{2},\) given that if the axis passes through the center, \(I=\frac{1}{12} M \ell^{2}(\) Fig. 20 \(\mathrm{f}\) and \(\mathrm{g})\)

When bicycle and motorcycle riders "pop a wheelie," a large acceleration causes the bike's front wheel to leave the ground. Let \(M\) be the total mass of the bike-plus-rider system; let \(x\) and \(y\) be the horizontal and vertical distance of this system's cM from the rear wheel's point of contact with the ground (Fig. \(72 ) .(a)\) Determine the horizontal acceleration \(a\) required to barely lift the bike's front wheel off of the ground. \((b)\) To minimize the acceleration necessary to pop a wheelie, should \(x\) be made as small or as large as possible? How about \(y ?\) How should a rider position his or her body on the bike in order to achieve these optimal values for \(x\) and \(y ?\) (c) If \(x=35 \mathrm{cm}\) and \(y=95 \mathrm{cm},\) find \(a\) .

(II) A merry-go-round accelerates from rest to 0.68 \(\mathrm{rad} / \mathrm{s}\) in 24 \(\mathrm{s}\) . Assuming the merry-go-round is a uniform disk of radius 7.0 \(\mathrm{m}\) and mass \(31,000 \mathrm{kg}\) , calculate the net torque required to accelerate it.

(II) A thin 7.0 -kg wheel of radius 32 \(\mathrm{cm}\) is weighted to one side by a 1.50 -kg weight, small in size, placed 22 \(\mathrm{cm}\) from the center of the wheel. Calculate \((a)\) the position of the center of mass of the weighted wheel and (b) the moment of inertia about an axis through its CM, perpendicular to its face.
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