2. Why can't an acrobat stop himself from
spinning while he is in midair?
When the acrobat is spinning in midair, he would need to exert a
non-zero net torque to decrease his angular momentum and stop
spinning. But when he is in midair, any force that he exerts
would also involve an equal and opposite reaction force according
to Newton's 3rd law. So, he cannot supply a non-zero net force
or torque.
4. Falling into a leaf pile is much more
comfortable than falling onto the bare ground. In both cases you
come to a complete stop, so why does the leaf pile feel so much
better?
Coming to a stop in both cases means that the change in linear
momentum is the same, but the force you experience while stopping
is not the same. The stopping force you feel in the pile of
leaves is less than that for bare ground, since the stopping time
is greater for the pile of leaves. The stopping time and stopping
force factors vary inversely with each other to keep the value for
the change in momentum the same in both cases. Recall that the
force multiplied by the time duration equals the momentum change
according to the proper form of Newton's 2nd law.
6. A horse does work on a cart it's pulling
along a straight, level road at a constant speed. The horse is
transferring energy to the cart, so why doesn't the cart go
faster and faster? Where is the energy going?
The horse is pulling on the cart with a non-zero net force, so it
is doing work on the cart. This positive work will increase the
kinetic energy of the cart, causing it to go faster if no other
force is at work. But, we also need to include the effects of
friction forces on the cart! Friction between the wheel hubs and
the cart's axles also do work on the cart, but in this instance
it is negative work since friction always opposes the motion of
an object. The work done by the horse is needed to offset the
negative work done by friction and keep the cart moving at
constant speed. If the horse did not put exactly as much energy
into the cart as friction removed into heat, the cart would slow
down and eventually stop.
10. If you sit in a good swivel chair with
your feet off the floor, the chair will turn slightly as you move
about but will immediately stop moving when you do. Why can't
you make the chair spin without touching something?
When you are sitting still in a good swivel chair, your momentum
is zero. If your feet are off the ground and you touch nothing
else, you cannot cause any external forces to act on you and the
chair. So your momentum remains zero, according to Newton's
laws. Now, if you twist or turn in the chair you are altering
your angular momentum and the chair compensates by turning in the
opposite direction to cancel your angular momentum. Again the
total angular momentum remains zero. The chair will move only
when you are moving in a way that keeps zero angular momentum,
and it will stop when you stop because then both you and the
chair have zero angular momentum individually. However, if your
feet touch the floor and you push on the floor, then the Newton's
3rd law reaction force will act as an outside force on both you
and the chair - permitting you to spin the chair.
12. Firefighters slide down a pole to get to
their trucks quickly. What happens to their gravitational
potential energy and how does it depend on the slipperiness of
the pole?
A firefighter at the top of the pole has more gravitational
potential energy than the same firefighter standing at the bottom
of the pole by an amount equal to the firefighter's weight
multiplying the change in height. As the firefighter slides down
the pole, some of this potential energy becomes the kinetic
energy of the firefighter's motion, in proportion to the distance
dropped so far. If the pole were perfectly slippery (or
frictionless), all of the incremental decrease in gravitational
potential energy would become the firefighter's kinetic energy.
The slipperiness of the pole depends on the friction that the
firefighter experiences when sliding down it. This friction
changes part of the decrease in gravitational potential energy
into heat, leaving the rest to be the firefighter's kinetic
energy.
16. As you begin pedaling your bicycle and
it accelerates forward, what is exerting the forward force that
the bicycle needs to accelerate?
The force accelerating the bicycle is the static friction force
at the contact regions of the bicycle tires with the ground. When
you pedal a bicycle, you exert a torque causing the tires to rotate
forward - driving the leading edge of the tire downward and toward
the rear of the bicycle. If there were no friction with the ground,
the bicycle tire would just spin and the bicycle would not move.
With friction, though, the force of static friction (when the tire
is not slipping or skidding) opposes the torque-derived force of the
bicycle tire trying to slide backward against it with an equal,
forward force. It is this forward-directed friction force that
accelerates the bicycle forward.
As long as you are content to travel straight ahead at constant
speed, Newton's laws tell us that inertia keeps the motion
steady. Any change to this motion, whether it be a turn or an
acceleration, requires the effective application of some force.
On an icy road, the ice reduces the friction between the tires
and the road. It is the friction forces, opposing the forces
applied by the contacting tires, that causes an acceleration of
the vehicle in turning or changing speed. So, we primarily see
that a problem exists when we try to use friction in some type
of acceleration.
26. When you first let go of a bowling ball,
it's not rotating. But as it slides down the alley, it begins to
rotate. Use the concept of energy to explain why the ball's
forward speed decreases as it begins to spin.
The bowling ball initially slides down the alley with a certain
amount of kinetic energy, determined by its mass and speed. If
the alley is level, its gravitational potential energy remains
constant. Whatever friction exists between the bowling ball and
the alley can only act to decrease the ball's energy, not add to
it. So, any change or redistribution of energy of the bowling ball
must effect the kinetic energy related to its forward motion. As
the ball begins to rotate, some kinetic energy is required for the
spin of the ball, depending on its moment of inertia (rotational
mass) and angular speed. The kinetic energy for rotation must come
from the available kinetic energy of forward, linear motion. Thus,
as the ball begins to spin, its forward kinetic energy is reduced.
Since its mass stays the same, its forward speed must decrease.
# 2
