Chap 1 Exercises

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Exercise 1 Imagine a robot like that shown in Chapter 1 but with only one finger instead of the two fingers that can move closer and farther from each other. How many degrees of freedom would the imagined robot have?

1       2       3       4       5       6       6      

question id: robot-finger-a

Suppose that the robot seen in the video in the chapter were able to move along the floor. How many degrees of freedom would it have then?

6       7       8       9      

question id: robot-finger-b

A roomba is a robotic cleaner for floors. How many degrees of freedom does it have? Give each of them descriptive names.

question id: roomba-df

Exercise 2 Which of these statements is correct about how a quantity differs from a number?

A number doesn’t have units.

A quantity is always positive.

A number is infinitely precise.

question id: quant-number

Exercise 3 What will be the units of a conversion factor from …

  1. km to meters
no units       km       km / m       m / km      

question id: convert-1

  1. from liters (say, of gasoline) to km
no units       can’t be done       km / liter       liter / km      

question id: convert-2

  1. from GBP (currency) to USD
no units       USD / euro       GBP / USD       USD / GBP      

question id: convert-3

  1. from miles-per-hour to distance in miles
can’t be done       hours       hours per mile       miles per hour      

question id: convert-4

Exercise 4 There are about 60 km in 100 miles. What is the corresponding conversion factor from km to miles? (Make sure to give units.)

question id: km-to-miles

Exercise 5 How could you specify the difference between two points on a 2-dimensional space such as a table-top? Make sure to give appropriate units.

question id: 2-d-difference

Exercise 6 How could you specify the difference between two points in a 3-dimensional space such as a room? Make sure to give appropriate units.

question id: 3-d-difference

Exercise 7 The following is a map of temperature during a summer day in the US.

You can also think of it as a function. If temperature is the output from the function, what is the input? Also, what is the dimension of the input space, that is, the “domain?”

question id: temp-map

Exercise 8 For each of the following, say whether the quantity is a pure number or if some kind of physical quantity is being represented.

  1. 17.3
Pure number       Time       Length       Mass       Temperature       Weight      

question id: YU5NCD-a

  1. \(\pi\)
Pure number       Time       Length       Mass       Temperature       Weight      

question id: YU5NCD-b

  1. 9 feet
Pure number       Time       Length       Mass       Temperature       Weight      

question id: YU5NCD-c

  1. 8 \(\mu\)m
Pure number       Time       Length       Mass       Temperature       Weight      

question id: YU5NCD-d

  1. 12.2 gm
Pure number       Time       Length       Mass       Temperature       Weight      

question id: YU5NCD-e

  1. 37\(^\circ\) C
Pure number       Time       Length       Mass       Temperature       Weight      

question id: YU5NCD-f

  1. 43 sec
Pure number       Time       Length       Mass       Temperature       Weight      

question id: YU5NCD-g

On paper

Exercise 9  

  1. Demonstrate that between any two integers \(a\) and \(b\) there is a rational number. (Hint: Do some arithmetic with \(a\) and \(b\) to construct such a rational number.)

  2. Demonstrate that between any two rational numbers \(a/b\) and \(c/d\) there is another rational number. (Hint: Construct such a rational number by arithmetic.)

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