Download Falling body problems

Falling body problems
Falling Body Problem 1
• A body weighing 8 lbs falls from rest toward the earth
from a great height. As it fall, air resistance acts upon
it, and it is assumed to be equal to 2v, where v is the
velocity ( in feet per second). Find the velocity and
distance fallen at time t seconds.
F = ma or ma = F
What is the mass?
What is shall be substituted in for a if you are going to try
to find velocity first?
What forces are acting on the body.
Known relationship
• Newton’s second law: F = ma where m =
w/g ; When Force is in lbs, then g = 32 ft/sec2,
in newtons, then 9.8 m/sec2, in dynes cm/sec2
• If y is the distance traveled, then the velocity
is v = dy/dt and acceleration is a = d2y/dt2
• Newton’s second law can be adapted to
F = m( dv/dt) or F = m(d2y/dt2)
Falling body problem 2
• A sky diver equipped with parachute falls from rest
towards the earth. The total weight of the man and
the equipment is 160 lbs. Before the parachute opens,
the air resistance (in pounds) is numerically equal to
(1/2)v, where v is the velocity ( in feet per second). The
parachute opens 5 sec after the fall begins, after it
opens the air resistance is numerically equal to (5/8)v2.
Find the velocity of the skydiver (a) before the
parachute opens and (b) after the parachute opens.
Exponential growth
• dx/dt represents the rate of change
• dx/dt = Kx is a formula which states that
decay (or growth) occurs at a rate that is
proportional to the amount present
• Solve that DE.
• If the amount at time t=0 is x0, then the
formula changes to
Problem 1
• Assume that the rate at which radioactive nuclei decay
is proportional to the number of such nuclei that are
present in a given sample. In a certain sample 10% of
the original number of radioactive nuclei have
undergone disintegration in a period of 100 years.
A. What percentage will remain after 1000 years.
B. In how many years will only one fourth of the
original number remain?
Mass on a Spring
Forces
• Force of gravity - mg
• Restoring force - (-kx – mg ) when k is the
spring constant as defined by Hooke’s law,
F
= kx
• Resisting Force (damping force)- a dx
dt
( if mass is moving downward this quantity is
negative)
• External impressed forces – F(t)
2
d
x
dx
and
m
a
kx F (t )
dt 2
dt
Free, Undamped Motion
• An 8 – lb weight is placed upon the lower end of
a coil spring suspended from the ceiling. The
weight comes to rest in its equilibrium position,
thereby stretching the spring 6 inches. The
weight is then pulled down 3 inches below its
equilibrium position and released at t – 0 with an
initial velocity of 1ft/sec, directed downward.
Neglecting the resistance of the medium and
assuming that no external forces are present,
determine the amplitude, period and frequency
of the resulting motion.
Free, Damped Motion
1. A 32-lb weight is attached to the lower end of a
coil spring suspended form the ceiling. The
weight comes to rest in its equilibrium, thereby
stretching the spring 2 ft. The weight is pulled
down 6 in below its equilibrium position and
released at t = 0. No external forces are present;
but the resistance of the medium in pounds is
numerically equal to 4 (dx/dt), where dx/dt is
the instantaneous velocity in feet per second.
Determine the resulting motion of the weight on
the spring.
Forced Motion
• A 16-lb weight is attached to the lower end of
a coil suspended from the ceiling, the spring
constant of the spring being 10ft/ft. The
weight comes to rest in its equilibrium
position. Beginning at t = 0 and an external
force given by F(t) = 5 cos 2t is applied to the
system. Determine the resulting motion if the
damping force is numerically equal to 2
(dx/dt), where dx/dt is the instantaneous
velocity in feet in second.