2. As you wind the mainspring of a mechanical
watch or clock, why does the knob get harder and harder to
turn?
Winding the mainspring of a mechanical watch is analogous to
stretching a linear spring further and further. As you continue,
the force you need to apply to stretch the spring increases, as
the spring force depends on the net distance stretched from the
spring's equilibrium length. The amount of work done to wind the
mainspring through equivalent amounts increases with the spring
force, as you continue to wind.
4. You take two boxes of cookies with you on
your vacation to the moon. One of the boxes is labeled in terms
of mass in kilograms, while the other is labeled in terms of
weight in pounds. When you arrive at the moon, where gravity is
much weaker than on earth, you find that one of the boxes is no
longer accurately labeled. Which label is now wrong and what has
caused the inaccuracy?
Provided that you have not eaten any cookies along the way, the
box labeled in pounds is incorrect on the moon. The box of
cookies labeled in kilograms tells the mass of the cookies
inside, which has not changed. But the box labeled in pounds is
giving the weight of the cookies, that is the force of gravity on
the mass of cookies. Since the acceleration due to gravity is
much less on the surface of the moon than on the surface of the
earth, this weight on the label is now wrong by a factor of the
ratio of the moon's gravitational acceleration to the earth's
gravitational acceleration.
6. Why are there no spring scales in which
the basket moves upward as you fill it with objects?
The springs in a spring scale weigh items by measuring the length
change in a spring needed to create a spring force equal to the
object's weight. The measure is taken when the net force on the
item is zero, when the downward force of gravity on the item is
exactly countered by the upward spring force. Since the force of
gravity on something is always directed downward, that is the
direction that the basket must move as it changes the length of
the spring.
12. People often remark that a particular
scale "reads heavy," meaning that it reports more than a person's
real weight. What is wrong with the scale's spring?
A scale reads out an object's weight by measuring the length
change of the spring in it, which should be linear with the
object's weight. For a scale to read 'heavier' than it should,
the scale's spring must be changing in length more than would be
expected for the given scale. This indicates that the spring is
not as stiff as it should be for the scale's calibration, causing
it to change more in length to compensate. So, the spring
constant (or Hooke's Law constant) of the spring is smaller than
it should be.
14. While you're weighing yourself on a
bathroom scale, you reach out and push downward on a nearby
table. Is the weight reported by the scale high, low, or
correct?
The scale measures your weight by exerting an upward force on you
exactly equal to your weight, yielding zero net force on you when
you are standing still. If you reach over and push downward on a
nearby table, you are supporting part of your weight on that
table. As you push down, Newton's 3rd law tells us that there
will be an equal, upward force. Since the total force downward
on you is your weight and part of the upward, normal force is
supplied by the neighboring table, the scale you are standing on
reports a low weight.
18. To weigh an infant you can step on a
scale once with the infant and then again without the infant.
Why is the difference between the scale's two readings equal to
the weight of the infant?
When you step on a scale with an infant, the spring in the scale
changes length to exactly compensate for your combined weights.
Then, when you step on again by yourself, it has a new, smaller
length change that only compensates for your weight. The
difference in length changes has a certain value corresponding to
some spring force. Since the length of a spring changes linearly
with applied force (weight), the difference in length changes
multiplied by the spring constant corresponds exactly to the
weight of the infant. (Arithmetically, this involves the
distributive property for multiplication - applied to Hooke's
Law.)
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