SP221/2141&4341
Chapter 5 Homework
Due Friday October 7, 2016
Question 1: A box of mass m1 =5.0 kg sits on a rough surface that is inclined at θ=31◦ . The
coefficient of static friction between the box and the surface is µs =0.22. The box is attached to
an ideal (massless, unstretchable) rope that runs over an ideal (massless, frictionless) pulley to a
hanging second block, of mass m2 . What is the range of masses m2 such that the blocks will remain
stationary? [10 points]
.0
=5
m1
kg
m2
.22
0
µ s=
31◦
Question 2: A small block of mass m is sliding around in a horizontal circle on the inside surface of
a frictionless cone whose side makes an angle θ with its center. The block is moving with a constant
speed v, and the horizontal circle it sweeps out is located at a height H above the point of the cone.
(a) Draw a free body diagram for the block, and label the direction of its acceleration. (b) Find
the magnitude of the normal force acting on the block in terms of the given quantities and g. (c)
What is the value for the constant speed v such that the block will continue to slide in the horizontal
circle with not friction. This answer should be independent of θ, so do any trigonometry needed to
eliminate it from your answer. [10 points]
θ
H
SP221/2141&4341
Chapter 5 Homework
Due Friday October 7, 2016
Question 3: A thin sheet uniform mass density consists of a circle or radius R in which a smaller
circle of radius R/2 has been cut out stretching form the center of the big circle to its edge. Where
is the center of mass of this object? We know it will be along the symmetry axis, so just give your
answer as the distance between the center of the big circle and the center of mass. (Hint: You can
treat this object as the superposition of two circles, the small of which has a negative mass but a
surface mass density of the same magnitude as that of the larger circle.) [5 points]
R
Question 4: A right circular cone of radius R and height H is made of a material with a uniform
mass density ρ. Where is the center of mass for this cone? We know by symmetry that it will be
along the central axis, so determine the height h of the center of mass above the circular base as
a function of H and R. (Hints: The volume of a right circular cone is given by V = πR2 H3 . You
should be able to get away with only doing an integral along the z-direction. You’ll just need to
determine a formula that give the radius of the circular cross sections as a function of z.) [15 points]
H
h
R
SP221/2141&4341
Chapter 5 Homework
Due Friday October 7, 2016
PROGRAMMING PROBLEM
Question 5: We have two identical bullets, each with a mass of 8.00 g. Each bullet starts at an
initial height of 2.00 m. Simultaneously, the first bullet is dropped while the second bullet is fired
horizontally at a speed of 380.0 m/s. In the absence
of air resistance, we know that the two bullets
q
2h0
=0.6386 s later). I have written a MATLAB
will hit the ground simultaneously (at t =
g
program that confirms this fact. What about in the presence of air resistance? Will they still hit
the ground at the same time? If not, which one hits first? (a) First, assume laminar or Stokes drag:
Fd = bv. (This model is typically valid for a slower moving object in a viscous fluid). Repeat the
calculation with Fd = bv and b=0.005 kg/s. How long does it take for each bullet to hit the ground?
(b) Second, assume turbulent or Newtonian drag: Fd = bv 2 . (This model is typically valid for a fast
moving object in a fluid with a low kinematic viscosity. This model is likely far more accurate for a
bullet fired in air.) Repeat the calculation with Fd = bv 2 and b=0.00006 kg/m. How long does it
take for each bullet to hit the ground? [20 points]
Note: Again, for air drag the magnitude is Fd = bv n and the direction points opposite the
velocity. It is NOT necessarily the case that |Fd,x | = b|vx |n . The component of the drag in the
x-direction may well depend on the component of the velocity in the y-direction and vice versa.