April 20, 1998

    The following text contains explanations of the most
frequently missed items on the Physics 201 examination #3
administered Thursday April 16, 1998.  The most commonly missed type
of question seems to be one involving use of ratios and proportions,
a simple math skill that many students in the class may want to
review and practice.  These are questions like: "If A = xy/z, by 
what factor does z change if A doubles, y decreases by 50% and x
stays the same?"
    Trigonometry and vector arithmetic also seems to remain
pretty difficult for many students.

1.  Since the momentum reverses direction, the CHANGE is
(.25 kg)(-40. m/sec)-(.25 kg)(+40. m/sec) = -20.0 kg-m/sec.
This is also the impulse delivered to the ball, which by
Newton's 3rd Law is the negative of the impulse delivered to
the bat.  Therefore I=20.0 N-s

6.  From conservation of mechanical energy, the speed of the
ball just before collison with the floor is square-root
of 2gh = 19.6, so mv = (0.5 kg)(4.43 m/s).  Because ball
reverses direction without change in speed, change in momentum
is twice that, or 4.43 kg-m/s.

7.  Vector subtraction of final and intitial momentums gives
change in momentum that has horizontal component of -3.0 kg-m/s
and vertical component of +4.0 kg-m/s.  From Pythagorean theorem
this gives magnitude 5.00 kg-m/sec.

9. Cylinder rolls without slipping, so distance travelled is
radius times angle of rotation, so
Angle in radians = distance/radius = 10 radians.

11.  The total acceleration has both centrifugal (v^2/R) and
tangential components, so its magnitude must be bigger than the
magnitude of the tangential component alone, i.e. > 0.314

13.  Use conservation of energy.  Kinetic energy at bottom equals
potential energy at top, which is mg(2m sin(30))=9.8 J. 

15.  Its angular speed at the bottom is (3.14 m/sec)/(0.1 m) =
31.4 rad/s, so angular acceleration = (31.4)/(1.3)=24.1 rad/s/s.

18.  From Stress = strain/Young's Modulus, the amount of stretch 
is directly proportional to the length and inversely proportional
to the cross-sectional area of the wire.  The cross section 
increases by factor of 4 and length only doubles, so stretch is
half its previous value, or 0.05 mm.

19.  The period is proportiuonal to the square root of the mass,
so doubling the period means the mass is four times greater.

20.  The potential energy of the spring is work done to stretch
it, and is proportional to the square of the amount of stretch.
Doubling the stretch quadruples the potential energy to 100 J, 
which is 75 J more than before.

21.  The period is 6.28 times square root of (l/g), independent of
mass and amplitude.

22.  Since angular momentum is conserved, moment of inertia must
decrease by 50% when angular speed doubles.  The rotational
kinetic energy doubles, and angular momentum doesn't change.
(Energy isn't conserved because it takes work to pull the 
weights inward.)

23.  The line of action of the weight passes through the center of
the bar and is directed downward, so moment arm is (0.5 m sin45) =
0.354 m, and weight is 98 newtons, so torque is 35 N-m.

24.  Moment arm of the tension force is double the moment arm of the
weight of the bar, so sum of torques is zero only if tension is
one-half of weight, or 49 N.

26.  Divide mass of 0.386 kg by density to get 2 millionth of a
cubic meter, which is 2.00 cubic centimeters.

28. The weight of displaced water was buoyant force, which,
from Newton's First Law, must be (9.8 - 8.2) N.  Ratio of
densities is ratio of weights (since V and g are the same for
both), so ratio is 9.8/1.6 = 6.125.

30. As explained in lecture, air speed over top of frizbee is greater,
so, by Bernoulli Principle, pressure is smaller there.

31. By Archimedes principle, since both displace the same volume of
fluid, the buoyant force is the same for both.  They fall at
different speeds (heavier ball bearing falls faster), so viscous
drag on ball bearing is bigger.

32.  The vector sum of weight and buoyant forces is proportional 
to the difference between density of material and density of fluid,
so ratio is (5500 - 1000)/(2500 - 1000) = 45/15 = 3.00 .

33.  Since density is ratio of mass to volume, and since neither of 
these change with temperature is the volume is held constant, density
of a constant volume of gas can't be used as a thermometer.  All other
items change with temperature.

36.  Buoyant force is weight of displaced air = (1.3)(9.8) N.

37.  98.6 F = 37 C = (37+273) K = 310 K

38.  Because weight of displaced water is weight of ice cube, 
which is the same as the weight of the melted ice, the melted 
ice has the same volume as the displaced water, and the level
of water doesn't change at all.

39.  Heat added is proportional to mass times temperature change.
Since heat added is half as much, and temperature change is 1/5
as much, mass must be 2.5 times greater.

40.  Heat added must be heat of fusion times mass of ice, or
60,300 J, so it would take 603 seconds at 100 J/s to melt the ice.
That's about 10 minutes.