Use the formulas for arc length and surface area and the
equation x^2+y^2=R^2
(a)... to derive the formula for the surface area of a sphere
with radius R
(b)... to derive the formula for the circumference of a circle
with radius R
Learning Goal:
To derive the formulas for the major characteristics of motion
as functions of time for a horizontal spring oscillator and to
practice using the obtained formulas by answering some basic
questions.
A block of mass m is attached to a spring whose spring constant
is k. The other end of the spring is fixed so that when the spring
is unstretched, the mass is located at x=0. (Figure 1). Assume that
the +x direction is to the right....
A body weighing 128 pounds hangs from a spring with constant
1400 lb/ft. The medium, where the body moves, offers a force of
opposition to the movement numerically equal to 15 for its
instantaneous speed. If the weight is released 2 feet above its
balance position, say how fast it should be initially pushed so
that after 6 seconds it reaches the lower limit position. Take the
constant of gravity as 32 ft/sec^2.
The length of nylon rope from which a mountain climber is
suspended has a force constant of 1.18 ✕ 104 N/m.
(a)
What is the frequency (in Hz) at which he bounces, given that
his mass plus the mass of his equipment is 78.0 kg?
(b)
How much would this rope stretch (in cm) to break the climber's
fall if he free-falls 2.00 m before the rope runs out of slack?
Hint: Use conservation of energy.
(c)
Repeat both parts...
The length of nylon rope from which a mountain climber is
suspended has a force constant of 1.40 104
N/m. (Hz)
(a) What is the frequency at which he bounces, given his mass plus
equipment to be 70.0 kg? (m)
(b) How much would this rope stretch to break the climber's fall,
if he free-falls 2.00 m before the rope runs out of slack?
(c) Repeat both parts of this problem in the situation where twice
this length of nylon rope...