/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 62 Responding to an alarm, a \(782-... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 62

Responding to an alarm, a \(782-\mathrm{N}\) firefighter slides down a pole to the ground floor, \(3.3 \mathrm{~m}\) below. The firefighter starts at rest and lands with a speed of \(4.2 \mathrm{~m} / \mathrm{s}\). Find the average force exerted on the firefighter by the pole.

Short Answer

Expert verified
The average force exerted on the firefighter by the pole is approximately 995.3 N.

Step by step solution

01

Identify Known Values

Let's list out the known values from the problem statement. The weight of the firefighter is given as 782 N, which implies mass can be calculated using gravitational acceleration \(g = 9.8 \text{ m/s}^2\). The initial velocity of the firefighter \(v_i\) is 0 m/s, the final velocity \(v_f\) is 4.2 m/s, and the distance \(d\) is 3.3 m.
02

Calculate Mass of Firefighter

The weight of the firefighter (782 N) is given by \(mg\), where \(m\) is the mass and \(g\) is the acceleration due to gravity. Thus, we can calculate the mass:\[ m = \frac{782}{9.8} \approx 79.8 \text{ kg} \]
03

Apply Kinematic Equation to Find Acceleration

Using the kinematic equation: \(v_f^2 = v_i^2 + 2ad\), where \(a\) is the acceleration, we solve for \(a\):\[ (4.2)^2 = 0 + 2a(3.3) \]\[ 17.64 = 6.6a \]\[ a \approx 2.67 \text{ m/s}^2 \]
04

Use Newton's Second Law

Newton's second law states \(F_{net} = ma\). The net force \(F_{net}\) is the force exerted by the pole and gravitational force together. So, \(F_{net} = F_{pole} - mg\), which gives \(F_{pole} = F_{net} + mg\). Substitute the known values: \[ F_{net} = m \cdot a = 79.8 \times 2.67 \approx 213.3 \text{ N} \] \[ F_{pole} = 213.3 + 782 = 995.3 \text{ N} \]
05

Conclude with the Average Force

The average force exerted on the firefighter by the pole is found to be approximately 995.3 N.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Kinematics
Kinematics is a branch of physics that deals with the motion of objects without considering the causes of that motion. It's a crucial tool for understanding and calculating various parameters like speed, velocity, acceleration, and displacement.

In this exercise, we need to determine how a firefighter's speed changes as they slide down a pole. Key components here include:
  • Initial Velocity (\( v_i \)): The velocity of the firefighter at the start is zero since they start from rest.
  • Final Velocity (\( v_f \)): After descending 3.3 meters, the firefighter reaches a velocity of 4.2 m/s.
  • Displacement (\( d \)): The firefighter moves 3.3 meters downwards.
  • Acceleration (\( a \)): We need to find this to calculate the net force.
To find the acceleration, we apply the kinematic formula: \[ v_f^2 = v_i^2 + 2ad \]This equation helps us solve for acceleration, revealing how quickly the velocity of the firefighter changes as they descend.
Gravitational Force
Gravitational force is the force of attraction between two objects with mass. On Earth, this force is what pulls objects towards the ground, commonly referred to as weight. It's calculated using the formula: \[ F_g = mg \]where \( m \) is the mass of the object and \( g \) is the gravitational acceleration, approximately \( 9.8 ext{ m/s}^2 \) on Earth.

For our firefighter:
  • Weight of the firefighter: Given as 782 N. This is the gravitational force exerted on the firefighter by Earth.
  • Mass Calculation: By rearranging the formula \( F_g = mg \), we find \( m = \frac{F_g}{g} \), allowing us to solve for the mass.
  • Mass Result: The firefighter鈥檚 mass is approximately 79.8 kg.
Understanding gravitational force helps link the changes in motion to the forces acting upon the object.
Force Calculation
Force calculation in this context involves Newton's Second Law of Motion. This law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration (\( F_{net} = ma \)). The firefighter experiences two main forces:
  • The gravitational force, pulling them downward.
  • The force exerted by the pole, which we need to find.
To discover the force exerted by the pole, you calculate:
  • First, determine the net force using \( m \times a \).
  • Net Force Calculation: Multiply the mass (79.8 kg) by the calculated acceleration (2.67 m/s虏) to find the net force around 213.3 N.
  • Add the gravitational force (782 N) to this net force to ensure the equation balances out because \( F_{pole} - mg = F_{net} \).
  • Force by the pole: This results in the pole exerting approximately 995.3 N on the firefighter.
This detailed approach in force calculation merges kinematics with dynamics, providing a comprehensive view of the forces at play during the firefighter's descent.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Analyze You throw a ball straight up into the air. It reaches a maximum height and returns to your hand. At what location(s) is the kinetic energy of the ball (a) a maximum and (b) a minimum? At what location(s) is the potential energy of the ball (c) a maximum and (d) a minimum?

Predict \& Explain You throw a ball upward and let it fall to the ground. Your friend drops an identical ball straight down to the ground from the same height. (a) Is the change in kinetic energy (from just after the ball is released until just before it hits the ground) of your ball greater than, less than, or equal to the change in kinetic energy of your friend's ball? (b) Choose the best explanation from among the following: A. Your friend's ball converts all of its initial energy into kinetic energy. B. Your ball is in the air longer, which results in a greater change in kinetic energy. C. The change in gravitational potential energy is the same for each ball, which means that the change in kinetic energy must also be the same.

Engine 1 produces twice the power of engine \(2 .\) If it takes engine 1 the time \(T\) to do the work \(W\), how long does it take engine 2 to do the work 3 W? Explain.

Analyze You throw a baseball glove straight upward to celebrate a victory. Its initial kinetic energy is \(K E\), and it reaches a maximum height \(h\). What is the kinetic energy of the glove when it is at the height \(h / 2\) ?

Connect to the Big Idea The mechanical energy of a falling ball stays the same, even though it is constantly speeding up. Similarly, if you throw a ball upward, it slows down, even though its mechanical energy stays the same. Explain.
See all solutions

Recommended explanations on Physics Textbooks

Further Mechanics and Thermal Physics

Read Explanation

Measurements

Read Explanation

Fluids

Read Explanation

Medical Physics

Read Explanation

Radiation

Read Explanation

Physical Quantities And Units

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.