/*! 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 84 A spectator, seated in the left-... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 26: Problem 84

A spectator, seated in the left-field stands, is watching a baseball player who is \(1.9 \mathrm{m}\) tall and is \(75 \mathrm{m}\) away. On a TV screen, located \(3.0 \mathrm{m}\) from a person watching the game at home, the image of this same player is 0.12 \(\mathrm{m}\) tall. Find the angular size of the player as seen by (a) the spectator watching the game live and (b) the TV vicwer. (c) To whom does the player appear to be larger?

Short Answer

Expert verified
(a) 0.0253 radians. (b) 0.04 radians. (c) The TV viewer sees the player as larger.

Step by step solution

01

Define the Problem

To find the angular size of the player as seen by the spectator and the TV viewer, we need to interpret the player's height and the distances involved in terms of angular size. Angular size can be found using the formula \( \theta = \tan^{-1} \left( \frac{h}{d} \right) \), where \( h \) is the height of the object and \( d \) is the distance from the observer to the object. We will have two scenarios to examine: the live viewer at the stadium and the viewer at home watching the TV.
02

Calculate the Angular Size for the Spectator

For the spectator, we use the player's height \( h = 1.9 \mathrm{m} \) and distance \( d = 75 \mathrm{m} \). The angular size \( \theta_1 \) is given by \( \theta_1 = \tan^{-1} \left( \frac{1.9}{75} \right) \). Calculating this gives:\[ \theta_1 = \tan^{-1} \left( 0.0253 \right) \approx 0.0253 \, \text{radians} \]
03

Calculate the Angular Size for the TV Viewer

For the TV viewer, the player is represented by a 0.12 \( \mathrm{m} \) tall image on the screen. The viewer is positioned 3.0 \( \mathrm{m} \) away from this TV. Thus, the angular size \( \theta_2 \) is calculated as \( \theta_2 = \tan^{-1} \left( \frac{0.12}{3.0} \right) \). Calculating this gives:\[ \theta_2 = \tan^{-1} \left( 0.04 \right) \approx 0.04 \, \text{radians} \]
04

Compare Angular Sizes

Now, we compare the two angular sizes. The angular size for the spectator watching live is \( 0.0253 \) radians, while for the TV viewer, it is \( 0.04 \) radians. Since \( 0.04 \) radians is greater than \( 0.0253 \) radians, the TV viewer sees the player with a larger angular size.

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.

Trigonometry in Physics
Trigonometry is incredibly useful in physics, especially when it comes to understanding how we perceive objects. When we talk about angular size, trigonometry helps us figure out how large an object appears based on its actual size and the distance from the observer. Angular size can be calculated using the tangent function from trigonometry. The formula is \( \theta = \tan^{-1} \left( \frac{h}{d} \right) \), where \( \theta \) is the angular size, \( h \) is the height of the object, and \( d \) is the distance to the object. This formula is derived from the basic trigonometric relationship in a right triangle. Using this formula, you can determine how much of your field of view an object takes up, which is critical in both everyday experiences and various applications in physics.
Angular Measurements
Angular measurements are a fundamental part of understanding the universe around us. They help us describe the position of stars in the sky, the orbit of planets, and even how a baseball player appears to an observer. In physics, angular size provides a measure of how "big" something appears based on its physical size and our distance from it. For instance, in the exercise provided:
  • The player appears to have different angular sizes to the live spectator and the TV viewer.
  • The height of the player and the distance each observer is at changes the angular measurement calculated.
While the player's physical height is constant, their angular size changes with distance, showcasing how angular measurements are crucial for understanding perception in physics.
Perception in Physics
Perception in physics often challenges our intuitive understanding of size and distance. Although a baseball player doesn't actually change size, how they appear to differ between observers due to varying angles and distances. This concept of perception helps explain why an object might seem larger on a screen than in real life, even if the screen is much closer to the viewer. In daily life:
  • Objects farther away appear smaller, an effect that can be quantified using angular size calculations.
  • Television screens can dramatize objects by making them appear larger, due to the decreased distance between the viewer and the image.
Understanding perception in physics allows us to explain and predict these variations, leading to applications in optics, astronomy, and even human vision theories.

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

A microscope for viewing blood cells has an objective with a focal length of \(0.50 \mathrm{cm}\) and an eyepiece with a focal length of \(2.5 \mathrm{cm} .\) The distance between the objective and eyepiece is \(14.0 \mathrm{cm} .\) If a blood cell subtends an angle of \(2.1 \times 10^{-5} \mathrm{rad}\) when viewed with the naked eye at a near point of \(25.0 \mathrm{cm},\) what angle (magnitude only) does it subtend when viewed through the microscope?

Violet light and red light travel through air and strike a block of plastic at the same angle of incidence. The angle of refraction is \(30.400^{\circ}\) for the violet light and \(31.200^{\circ}\) for the red light. The index of refraction for violet light in plastic is greater than that for red light by 0.0400. Delaying any rounding off of calculations until the very end, find the index of refraction for violet light in plastic.

Visitors at a science museum are invited to sit in a chair to the right of a full-length diverging lens \(\left(f_{1}=-3.00 \mathrm{m}\right)\) and observe a friend sitting in a second chair, \(2.00 \mathrm{m}\) to the left of the lens. The visitor then presses a button and a converging lens \(\left(f_{2}=+4.00 \mathrm{m}\right)\) rises from the floor to a position \(1.60 \mathrm{m}\) to the right of the diverging lens, allowing the visitor to view the friend through both lenses at once. Find (a) the magnification of the friend when viewed through the diverging lens only and (b) the overall magnification of the friend when viewed through both lenses. Be sure to include the algebraic signs \((+\) or \(-)\) with your answers.

The angular magnification of a telescope is 32800 times as large when you look through the correct end of the telescope as when you look through the wrong end. What is the angular magnification of the telescope?

A camper is trying to start a fire by focusing sunlight onto a piece of paper. The diameter of the sun is \(1.39 \times 10^{\circ} \mathrm{m},\) and its mean distance from the earth is \(1.50 \times 10^{11} \mathrm{m} .\) The camper is using a converging lens whose focal length is \(10.0 \mathrm{cm}\). (a) What is the area of the sun's image on the paper? (b) If \(0.530 \mathrm{W}\) of sunlight passes through the lens, what is the intensity of the sunlight at the paper?
See all solutions

Recommended explanations on Physics Textbooks

Fundamentals of Physics

Read Explanation

Linear Momentum

Read Explanation

Radiation

Read Explanation

Energy Physics

Read Explanation

Modern Physics

Read Explanation

Electricity and Magnetism

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.