HW #8 - Otterbein

FS 2013
Homework #8
(due Wednesday, October 23, 2013 at the beginning of class)
1. Forces and Potential energy:
a. Consider a spring with a force represented by F  (ky  py 3  qy 4 ) ˆj . Is this
force conservative? Explain why or why not. Determine the potential energy
function if possible.
b. Find the force (vector!) associated with the potential energy function U(x,y,z) =
4y2x +3x2+2xy.
c. In the fashion of Sec. 8-6, especially Fig. 8-9, discuss the following potential:
U(x)= –k2/x + L2/x2, where k and L are constants. Plot the potential and the force
as a function of the distance x qualitatively, find the points where they are
maximal and minimal, and discuss possible turning points. What is the difference
in motion of particles with positive versus negative mechanical energy? If it
helps, choose both constants to be one (units?).
2. A block of mass 1kg is sliding down a track consisting of a quarter of a circle of radius
2m which the block slides down on its concave side, followed by a straight horizontal
section. The track is set up as follows. The quadrant of the circle is frictionless. The
horizontal track has a coefficient of friction of 0.25 up to a point at which the C which is
the end of a relaxed, compressible spring of stiffness K. The track under the spring is
a. Draw the physical situation.
b. Assuming the block starts from rest, determine the velocity of the block at the
end of the circular part of the track.
c. Find the thermal energy produced as the block slides to C.
d. What is the velocity of the block just before it hits the spring?
e. Calculate the stiffness of the spring.
3. In the test phase of the now-retired space shuttle, a glider of mass 980 kg was used. The
glider did glide back to Earth after been released horizontally at an altitude of 3.5km at a
speed of 480km/h. The touchdown speed was measured to be 210 km/h.
a. Determine the touchdown speed if there hadn’t been any air resistance.
b. Find the average force of air resistance assuming the angle of decent was a
constant 12 degrees with respect to the surface.
4. Consider a ½ kg mass glued to the right side of a 60cm long spring of stiffness 40N/m
attached to a wall on the left. Initially the mass is at rest at the equilibrium point of the
spring on a frictionless surface. When a constant horizontal force is applied to the block,
it moves to the right along the surface.
a. Draw a diagram of the physical situation and the free-body diagram of the mass.
b. Draw the energy bar chart and dind the velocity of the block after it has moved
25cm to the right.
c. Once it reaches the 25cm mark, the constant force is removed. Draw another
energy bar chart to find out how close will it subsequently get to the wall to which
the spring is attached.
5. Find the center of mass of a system of 8 particles located at the corners of an imaginary
cube placed in the first octant of a three-dimensional coordinate frame with one corner at
the origin. Assume the cube’s side length is 2m, and the masses of the particles are
determined by squaring the sum of their coordinates in kilograms and adding 1 kg, e.g.
the particle farthest from the origin at (2m,2m,2m) )will have a mass (2+2+2)2 kg + 1kg=
6. Consider a bullet of mass 0.1kg and a speed of 600 m/s which penetrates a distance
0.15m into a block of wood that is mounted firmly to the ground. Assuming the force
stopping the bullet is constant, determine the following:
a. Acceleration of the bullet
b. Force on the bullet
c. Time it takes the bullet to stop
d. Impulse of the force
e. Ratio between the impulse and the initial momentum of the bullet
7. Anatomy of a car crash: Mr. Somebody’s truck (2300kg), stopped at a red light, was hit
from behind by a 930kg car. The bumpers locked and so were the brakes, as the truck and
the car skidded 2.8m before coming to a final stop. Estimating the friction coefficient
between tires and road to be 0.8, what was the speed of the offender at impact?
8. A radioactive nucleus makes a radioactive decay into a daughter nucleus, a neutrino and
an electron. Assume the latter two are emitted at right angles and have momenta of 6.2 x
10-23 kg m/s and 9.6 x 10-23 kg m/s, respectively. What is the direction and magnitude of
the momentum of the recoiling daughter nucleus?
9. Consider a projectile of mass 20kg that is fired with initial velocity of 250m/s at an angle
of 60 degrees above the horizontal. As it reaches its maximal altitude, it explodes into
two equal-mass parts. One of those falls to the ground vertically with vanishing initial
a. Calculate the range of the other part of projectile.
b. Determine the amount of energy released during the explosion.