Download 4.1 The Concepts of Force and Mass

6.3.1. An elevator supported by a single cable descends a shaft at a
constant speed. The only forces acting on the elevator are the
tension in the cable and the gravitational force. Which one of the
following statements is true?
a) The work done by the tension force is zero joules.
b) The net work done by the two forces is zero joules.
c) The work done by the gravitational force is zero joules.
d) The magnitude of the work done by the gravitational force is larger
than that done by the tension force.
e) The magnitude of the work done by the tension force is larger than
that done by the gravitational force.
6.3.5. Consider the box in the drawing. We can slide the box up the frictionless
incline from point A and to point C or we can slide it along the frictionless
horizontal surface from point A to point B and then lift it to point C. How does the
work done on the box along path A-C,WAC, compare to the work done on the box
along the two step path A-B-C, WABC?
a) WABC is much greater than WAC.
b) WABC is slightly greater than WAC.
c) WABC is much less than WAC.
d) WABC is slight less than WAC.
e) The work done in both cases is the same.
6.5.1. Two balls of equal size are dropped from the same height
from the roof of a building. One ball has twice the mass of the
other. When the balls reach the ground, how do the kinetic
energies of the two balls compare?
a) The lighter one has one fourth as much kinetic energy as the
other does.
b) The lighter one has one half as much kinetic energy as the
other does.
c) The lighter one has the same kinetic energy as the other does.
d) The lighter one has twice as much kinetic energy as the other
does.
e) The lighter one has four times as much kinetic energy as the
other does.
5.2.1. A ball is whirled on the end of a string in a horizontal circle of radius R at
constant speed v. By which one of the following means can the centripetal
acceleration of the ball be increased by a factor of two?
a) Keep the radius fixed and increase the period by a factor of two.
b) Keep the radius fixed and decrease the period by a factor of two.
c) Keep the speed fixed and increase the radius by a factor of two.
d) Keep the speed fixed and decrease the radius by a factor of two.
e) Keep the radius fixed and increase the speed by a factor of two.
5.3.3. A ball is attached to a string and whirled in a horizontal circle. The ball
is moving in uniform circular motion when the string separates from the
ball (the knot wasn’t very tight). Which one of the following statements
best describes the subsequent motion of the ball?
a) The ball immediately flies in the direction radially outward from the center
of the circular path the ball had been following.
b) The ball continues to follow the circular path for a short time, but then it
gradually falls away.
c) The ball gradually curves away from the circular path it had been
following.
d) The ball immediately follows a linear path away from, but not tangent to
the circular path it had been following.
e) The ball immediately follows a line that is tangent to the circular path the
ball had been following
5.3.5. Imagine you are swinging a bucket by the handle around in a circle that is
nearly level with the ground (a horizontal circle). What is the force, the physical
force, holding the bucket in a circular path?
a) the centripetal force
b) the centrifugal force
c) your hand on the handle
d) gravitational force
e) None of the above are correct.
Oscillatory motion
(non-constant acceleration)
Simple Harmonic
Motion
10.1 The Ideal Spring and Simple Harmonic Motion
Applied
x
F
 kx
spring constant
Units: N/m
10.1 The Ideal Spring and Simple Harmonic Motion
HOOKE’S LAW: RESTORING FORCE OF AN IDEAL SPRING
The restoring force on an ideal spring is
Fx  k x
 F  kx  ma
x
10.2 Simple Harmonic Motion and the Reference Circle
x  Acost


k
m
file:///Users/silvinagatica/Desktop/simulatio
ns/applets/sim08.htm
amplitude A: the maximum displacement
period T: the time required to complete one cycle
frequency f: the number of cycles per second (measured in Hz)
1
f 
T
2
  2 f 
T
10.2 Simple Harmonic Motion and the Reference Circle
DISPLACEMENT
x  Acost
vx  A sin t
VELOCITY

ACCELERATION

ax  A 2 cost
max  mA sin t  m Asin t  kx
2
2


k
m
m 2  k
10.2 Simple Harmonic Motion and the Reference Circle
FREQUENCY OF VIBRATION

k
m
T
T1  T2


f  1/T
m1  m2
1   2
2

10.2 Simple Harmonic Motion and the Reference Circle
Example 6 A Body Mass Measurement Device
The device consists of a spring-mounted chair in which the astronaut
sits. The spring has a spring constant of 606 N/m and the mass of
the chair is 12.0 kg. The measured
period is 2.41 s. Find the mass of the
astronaut.
10.2 Simple Harmonic Motion and the Reference Circle
k

mtotal
mtotal  k  2
  2 f 
mtotal 
mastro
k
 mchair  mastro
2
2 T 
k

 mchair
2
2 T 
2

606 N m 2.41 s 

 12.0 kg  77.2 kg
4 2
2
T
10.3 Energy and Simple Harmonic Motion
F=-kx
F
Work=Area = 12
x0
-kx0
xf
x f  (kx f )  12 x 0  (kx0 )
x
Welastic  12 kxo2  12 kx2f
-kxf

10.3 Energy and Simple Harmonic Motion
DEFINITION OF ELASTIC POTENTIAL ENERGY
The elastic potential energy is the energy that a spring
has by virtue of being stretched or compressed. For an
ideal spring, the elastic potential energy is
PEelastic  12 kx2
SI Unit of Elastic Potential Energy: joule (J)
10.3 Energy and Simple Harmonic Motion
E f  Eo
1
2

1
2
mv 2f  mgh f  12 ky2f  12 mvo2  mgho  12 kyo2
kho2  mgho
2mg
ho 
k
20.20 kg  9.8 m s 2

 0.14 m
28 N m

simulation

10.2 Simple Harmonic Motion and the Reference Circle
SUMMARY:
x  Acost
T
Period T: time when ωt = 2 π
ω: angular frequency (how fast it ocillates)
4m
4k
 /2
2
2T
T /2


simulation horizontal oscilator



f  1/T
Frequency f: # cycles per second

2
k
m
10.4 The Pendulum
A simple pendulum consists of
a particle attached to a frictionless
pivot by a cable of negligible mass.
Does T depend on the mass?
simulation
10.4 The Pendulum
A simple pendulum consists of
a particle attached to a frictionless
pivot by a cable of negligible mass.

g
L
  2 f 
(small angles only)
2
g

T
L
T  2

simulation
L
g
How much would you change L to double
T?
10.4 The Pendulum
Example 10 Keeping Time
Determine the length of a simple pendulum that will
swing back and forth in simple harmonic motion with
a period of 1.00 s.
2
  2 f 

T
g
L
T 2g
L
4 2


T 2 g 1.00 s  9.80 m s 2
L

 0.248 m
2
2
4
4
2
10.1 The Ideal Spring and Simple Harmonic Motion
Example 1 A Tire Pressure Gauge
The spring constant of the spring
is 320 N/m and the bar indicator
extends 2.0 cm. What force does the
air in the tire apply to the spring?
10.1 The Ideal Spring and Simple Harmonic Motion
Applied
x
F
 kx
 320 N m 0.020 m   6.4 N
10.5 Damped Harmonic Motion
In simple harmonic motion, an object oscillated
with a constant amplitude.
In reality, friction or some other energy
dissipating mechanism is always present
and the amplitude decreases as time
passes.
This is referred to as damped harmonic
motion.
10.5 Damped Harmonic Motion
1) simple harmonic motion
2&3) underdamped
4) critically damped
5) overdamped
10.6 Driven Harmonic Motion and Resonance
When a force is applied to an oscillating system at all times,
the result is driven harmonic motion.
Here, the driving force has the same frequency as the
spring system and always points in the direction of the
object’s velocity.
10.6 Driven Harmonic Motion and Resonance
RESONANCE
Resonance is the condition in which a time-dependent force can transmit
large amounts of energy to an oscillating object, leading to a large amplitude
motion.
Resonance occurs when the frequency of the force matches a natural
frequency at which the object will oscillate.