Problem set 2, due Monday October 27.

There are 4 cases containing a total of 22 parts, each worth 4 points. The two exercises together are worth 8 points.

You will receive 4 more points for writing your name neatly and legibly on the finished problem set.

 

 

 

 

Problem 1:

Chapter 2, Case 1, all parts (p. 118)

A spring bathroom scale…

  1. As the scale slows you to rest, you are accelerating upward, so the scale exerts a force larger than your weight. (When you eventually come to rest, only then will the scale exert a force exactly equal to your weight).
  2. If one foot is in the air, and the other on the scale, the scale will report exactly your weight. It doesn't matter how much surface area is touching the scale, because the upward force from the scale is the same as when you stand on two feet.
  3. As you go from zero velocity to some positive upward velocity, you are accelerating upward, so the scale reads greater than your weight.
  4. The sum of the two readings will be equal to your weight, but not necessarily half-and-half (depends how you shift your body).
  5. The top scale will read your weight, the bottom scale your weight plus 10 Newtons.

 

Problem 2:

Chapter 3, Case 5, all parts (p. 155)

(Blimps are compact versions…)

  1. Since the volume of the blimp doesn't change, the upward buoyancy force from the air is constant. But the weight of the balloon increases as air is pumped in. ("less buoyant" is confusing here--what is happening is that the downward force from gravity is increasing relative to the upward buoyancy force from the air).
  2. pumping air to the rear causes a net torque about the center of mass, so the blimp tilts
  3. In warmer air, there is a smaller buoyancy force (see "check your understanding", pg. 130, answer at end of that section). So you should remove air from the ballonets, in order to decrease the weight of the balloon.
  4. In the process of kicking the air backwards, the fan exerts a force on the air. The air exerts a force on the fan in the opposite direction, which pushes the fan (and the blimp) forward.

 Problem 3:

Chapter 3, Case 12, all parts (p. 156)

(Some banks use pneumatic tubes…)

  1. The air pressure behind must be greater than the air pressure in front. That way there will be a net force in the forward direction, which will accelerate the tube forward.
  2. To maintain constant velocity, the net force must be zero. Since the tube is horizontal, the only horizontal forces are from the air pressure, which therefore must be equal back and front.
  3. Now the tube is vertical. The air pressure below must be greater than above, in order to balance the weight of the tube.
  4. The tube is again level, but is slowing down--in other words accelerating in the backward direction (even though it's MOVING forward!). Since there must be a net force to the rear, the air pressure in front is greater than behind.
  5. If you want to propel the cylinder forward, apply the low pressure to the front of the tube.
  6. Again, assuming you want to propel the cylinder forward, apply the high pressure to the back of the tube.

Problem 4:

Chapter 4, Case 7, all parts (p. 207)

(A kite is an airfoil…)

  1. At a given height, pressure and velocity are inversely related: as velocity decreases, pressure increases. Since the air is at atmospheric pressure before it hits the kite, the pressure becomes higher than atmospheric as the air slows down under the lower surface.
  2. The pressure above the kite will be somewhat lower than atmospheric pressure, depending on how much the air separates from the surface. (Imagine that the kite became so gigantic that it was like a wall. Then the air pressure on the downstream side of the wall would be just atmospheric pressure.)
  3. The overall force from the air is partly upward (to balance gravity) and partly horizontal in the direction away from the string (to balance the tension in the string.) In other words, the force from the air is angled upwards.
  4. If the upper surface is bowed, the air would flow faster on top and the pressure would decrease there. So there would be a bigger pressure difference between top and bottom. Another way to think about this is that in order to get the air to push you up, you must deflect it down. If the air on the top surface doesn't separate, you can deflect it down a lot better.

    5.  

  5. Because of the symmetry, there's no difference to the kite if it's upside down or not.
  6. If the kite starts to rotate, the weight of the tail exerts a torque about the center of mass that causes the kite to rotate back to its original position.
  7. If the tail is blown out away from the kite, then as the kite tries to rotate the drag on the tail will exert a torque tending to keep the tail behind the kite (away from the side with the string).

Problem 5:

Exercises 24 and 25, page 181. Answer these fairly completely, as a pair.

 Both of these questions pertain to terminal velocity, which is the velocity at which air drag is exactly equal to weight. At that point you will no longer accelerate.

As you get smaller, your weight decreases faster than your surface area. (Imagine a cube. If each side goes to half its original length, the volume goes to one-eighth, but a face of the cube only goes to one-fourth.) Since air drag depends on surface area, and your surface area is relatively bigger than your weight when you're small, your terminal velocity will decrease. So you'll hit the ground going SLOWER and you also WEIGH LESS, so the ground exerts a SMALLER FORCE on you to stop you, which are all good things.

The only bad thing is that your bones get weaker as you get smaller, but they get smaller in the same way that your surface area does. That is, your bone strength doesn't decrease as fast as your weight when you get smaller.