2. Why can't you open a door by pushing its
doorknob directly toward or away from its hinges?
When you exert a force directly toward or away from the hinges,
you are not exerting a net torque on the door to make it rotate
about its hinges. The torque depends on the size of the force
exerted at a right angle to the door (or more accurately, at a
right angle to the distance vector coming straight out from the
rotation axis of the hinges to the force vector, which is then
multiplied by the length of the distance vector). Here, no
component of the force vector is at a right angle to the door
since it is parallel to the door. This gives a net torque of
zero, and the applied force only attempts to push the door
straight into the hinges or to tear it off of the hinges.
6. The vast majority of wood screws have
"right-handed" threads, meaning that each screw moves in the
direction of its angular velocity as defined by the "right-hand
rule." For example, if a screw's angular velocity points into
the wood, the screw will move into the wood. If you are using
such a screw to hold down a loose floorboard, in which direction
should you turn the screwdriver (a tool for exerting torques on
screws) so that the screw will move into the floor?
From your perspective above (or behind) the wood screw, the
angular velocity of the screw will be into the floor if the
screwdriver is turned clockwise. Think about curling the fingers
of your right hand while your thumb points out at a right angle
from your hand, along the axis of your curled fingers and toward
the floor.
10. It's much easier to carry a weight in
your hand when your arm is at your side than it is when your arm
is pointing straight out in front of you. Use the concept of
torque to explain this effect.
When your arm is extended downward at your side, the weight is
also acting downward, in just about the same direction as your
arm. The torque on your shoulder due to the object in this
position is nearly zero since the direction of the weight is
almost parallel with your arm, which forms the distance vector.
(The weight alone must be additionally supported by your shoulder
without further complication by torque.) When your arm is
pointing straight out in front of you, the weight still acts
vertically downward at a right angle to your arm. So the torque
produced on your shoulder is about as large as it can be for the
given weight, making it much harder to carry.
14. How does the string of a yo-yo get the
yo-yo spinning?
The string of a yo-yo applies a force on the yo-yo that is the
tension in the string. This force acts at some distance from the
center of rotation of the yo-yo and about at right angles to this
distance vector, producing a torque on the yo-yo. This torque
then makes the yo-yo spin.
16. How does a bottle opener use mechanical
advantage to pry the top off a soda bottle?
A bottle opener acts as a lever with its fulcrum, or rotation
axis, located at the end touching the bottle top at its farthest
point. A small tab underneath the opener catches the lower lip
of the bottle top and can transmit a force there at some small
distance from the fulcrum. You apply an upward force on the
opener at a greater distance from the fulcrum, exerting a torque
on the top given by the size of the force multiplying the
distance from the fulcrum. Newton's laws tell us that as long as
the top does not come off, the top must be exerting an equal and
opposite torque on the opener. As you increase the upward force
you apply, the counter-torque of the bottle top must increase to
match. Eventually, the matching torque required of the bottle
top is greater than its mechanical strength and it comes off,
possibly bending as it does so. The actual size of the force at
the location of the bottle top's edge is greater than your
applied force by a multiplicative factor of your active lever
distance divided by the opener tab's lever distance. This is
called the mechanical advantage.
20. The basket of a wheelbarrow is located
in between its wheel and its handles. How does this arrangement
make it relatively easy for you to lift a heavy load in the
basket?
The force that must be exerted on a heavy load to lift and carry
it is its weight. When this load sits in the basket of a
wheelbarrow, we can use a lever's mechanical advantage to make
the job easier. The fulcrum of a wheelbarrow's "lever" is
located at its wheel. A force exerted at the far end of its
handles translates into a greater force at the location of the
basket when considered via torque. The load's weight produces a
torque on the wheelbarrow given approximately by the weight
multiplying the distance of the basket from the front wheel.
Lifting this load will require a counter-torque of about equal
size. But the torque produced at the handle end is the product
of the force exerted and the greater distance of this end from
the front wheel. Since the torques are equal in size, the
lifting force needed is smaller than the load's weight by the
ratio of the wheel-basket distance to the wheel-handle
distance.
22. When an airplane starts its propellors,
they spin slowly at first and gradually pick up speed. Why does
it take so long for them to reach their full rotational
speed?
The engines of an airplane can produce a certain amount of torque
to make their attached propellors spin. Since the propellors
were initially at rest, this torque causes them to have an
angular acceleration given by the torque divided by the
propellor's moment of inertia (or rotational mass). So the
propellors pick up angular speed at the rate given by the angular
acceleration, just as happens for motion in a straight line.
Eventually, they reach their full rotational speed when the
torque supplied by the engines is countered by the torques
arising from friction and air resistance, and their angular
acceleration becomes zero.
28. An automobile wheel has an air inlet
nipple. Why is a small weight usually added to the wheel on the
opposite side of the wheel from the nipple?
Many of us know from everyday life that an "unbalanced"
automobile wheel shakes or shudders when it is rotating at higher
speeds. An unbalanced wheel actually means that the mass of the
wheel and items attached to it are not uniformly distributed
around the wheel. The air inlet nipple is one example of
such an unbalanced mass on a wheel. This unbalanced mass
produces a torque on the rotating wheel that reverses direction
as it goes from the leading side of the tire to the trailing
side. The alternating torque attempts to make the wheel rotate
in alternating directions and so causes it to shake from side to
side. If a second, balancing mass is added to the wheel opposite
the unbalanced mass (along a wheel diameter), the paired masses
now produce equal and opposite torques so the wheel experiences
no angular acceleration to make it shudder.
30. The pedals of a bicycle are attached to
its crank. Suppose you stop pedaling your bicycle while the arms
of the crank are horizontal. Then one pedal will be in fornt of
the crank's center while the other pedal will be behind that
center. If you then push downward on both pedals at once, with
the same force on each, the crank won't undergo any angular
acceleration. Explain.
The pedals and bicycle crank are level and horizontal when equal
downward forces are applied to both pedals. The downward forces
on the pedals exert torques on the crank, but the torques cancel
each other since the distance vectors from the crank to each of
the pedals are in opposite directions. The force on the front
pedal tries to make the crank rotate toward the front of the
bicycle, while the one on the rear pedal tries to make the crank
rotate backward. Since the net torque is zero, the crank has no
angular acceleration.
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