Download lecture 3 pendulum and energy

LECTURE 3
PENDULUM AND ENERGY
Instructor: Kazumi Tolich
Lecture 3
2
¨
Reading chapter 13.5 – 13.6
Mechanical energy conservation in oscillatory motion
¤ Pendulum
¤
Simple pendulum
n Physical pendulum
n
Mechanical energy conservation
3
¨
¨
If non-conservative force (friction, air resistance etc) is not doing any work, the mechanical
energy is conserved.
The total mechanical energy of a mass on a spring is 𝐸 = 𝐾 + 𝑈 = &'𝑚𝑣 * + &'𝑘𝑥 * = &'𝑘𝐴* .
Quiz: 1
¨
¨
When does a body moving in simple harmonic motion have the maximum
acceleration? Choose all that apply.
When it has
A.
B.
C.
D.
E.
F.
G.
maximum velocity.
zero velocity.
maximum kinetic energy.
minimum kinetic energy.
maximum potential energy.
minimum potential energy.
zero displacement.
Quiz: 3-1 answer
¨
When it has
A.
B.
C.
D.
E.
F.
G.
¨
maximum velocity.
zero velocity.
maximum kinetic energy.
minimum kinetic energy.
maximum potential energy.
minimum potential energy.
zero displacement.
When the spring is maximally
stretched/compressed, the force on the mass is
maximum resulting in the maximum acceleration.
Quiz: 2
6
¨
A mass oscillates in simple harmonic motion with amplitude 𝐴. If
the mass is doubled, but the amplitude is not changed, what will
happen to the total mechanical energy of the mass-spring
system?
A.
B.
C.
Increases
Stays the same
Decreases
Quiz: 3-2 answer
7
¨
¨
¨
Stays the same
The total mechanical energy is equal to the initial value of the elastic
potential energy, which is 𝑈 = &'𝑘𝐴* .
This does not depend on mass, so a change in mass will not affect the
energy of the system.
Example: 1 (Walker Ch. 13-53)
8
¨
A block with a mass of 𝑚 = 0.505 kg slides
on a frictionless horizontal surface with a
speed of 𝑣 = 1.18 m/s. The block encounters
an unstretched spring and compresses it
𝐴 = 23.2 cm before coming to rest.
A.
B.
C.
What is the force constant of this spring?
For what length of time is the block in
contact with the spring before it comes to
rest?
If the force constant of the spring is
increased, does the time required to stop
the block increase, decrease, or stay the
same?
Simple pendulum
9
¨
¨
¨
The weight of the bob provides the restoring force.
Simple pendula do not exhibit true simple harmonic motion
for any angle. However, if the angle of oscillation 𝜃 is small,
the motion is close to and can be modeled as simple harmonic
motion.
The period of a simple pendulum with a length 𝐿, oscillating
with a small amplitude is given by
𝑇 = 2𝜋
𝐿
𝑔
Demo 1
10
¨
Simple Pendula with Different Lengths and Masses
¤ Demonstration
of the relationship between 𝐿 and 𝑇 : 𝑇 = 2𝜋
6
7
Quiz: 3
11
¨
The graph shows the square of the period versus the length of a simple pendulum on
a certain planet. What is the acceleration due to gravity on that planet in m/s2?
Quiz: 3-3 answer
12
¨
4 m/s2
¨
𝑇 = 2𝜋
6
7
¨
𝑔 = 2𝜋
* 6
8'
= 2𝜋
*
9 :
9; <'
= 4 m⁄s *
Quiz: 4
13
¨
Two pendula have identical periods. One has a slightly larger amplitude than the other, but
both swing through small angles compared to vertical. Which of the following must be true of
the pendulum that has the larger amplitude? Choose all the apply.
A.
B.
C.
D.
E.
F.
G.
H.
It has more mass than the other one.
It has less mass than the other one.
It is longer than the other one.
It is shorter than the other one.
It has slightly more energy than the other one.
It has slightly less energy than the other one.
It moves faster at the lowest point in its swing than the other one.
It moves slower at the lowest point in its swing than the other one.
Quiz: 3-4 answer
14
¨
It moves faster at the lowest point in its swing than the other one.
¨
𝑇 = 2𝜋
¨
¨
¨
¨
6
7
The period of the pendulum at a location depends only on the length,
independent of its amplitude or mass.
The two pendula must be of the same length.
The greater the amplitude, the greater distance that the bob must travel.
The energy of the pendulum depends on the mass of the bob, but we
have no information about it.
Physical pendulum
15
¨
¨
A physical pendulum is a rigid object free to rotate about a horizontal
axis that is not through its center of mass that oscillates when displaced
from equilibrium.
Physical pendula do not exhibit true simple harmonic motion for any
angle. However, if the angle of oscillation is small, the motion is close to
and can be modeled as simple harmonic motion with its period given by
𝑇 = 2𝜋
𝑙
𝑔
𝐼
𝑚𝑙 *
where 𝑙 is the distance between the axis and the center of mass, 𝐼 is the moment
of inertia about the axis, and 𝑚 is the mass of the object.
Demo: 2
16
¨
Physical pendulum
¤ Demonstration
C
of the relationship between 𝑇 and D : 𝑇 = 2𝜋
D
7
C
ED '
Quiz: 5
17
¨
Recall that period of oscillation of a physical pendulum is given by
𝑇 = 2𝜋
D
7
C
ED '
. The moment of inertia is always proportional to the
mass of the object 𝐼 ∝ 𝑚. Suppose that each of two objects is suspended
from a pivot point such that the distances between the center of mass and
the pivot point, 𝑙, is the same. Does it mean then that the periods of
oscillation must be the same?
A.
B.
Yes
No
Quiz: 3-5 answer
18
¨
¨
¨
¨
No
Let 𝐼 = 𝐶𝑚
Even if 𝑙’s are the same, if the objects are of the different shapes such that
𝐶’s are different, they would have different oscillation periods.
𝑇 = 2𝜋
D
7
C
ED '
= 2𝜋
C
7ED
= 2𝜋
HE
7ED
= 2𝜋
H
7D
Example: 2
19
¨
Consider the two pendula shown. In Case 1 a stick
of mass 𝑀 is pivoted at one end and used as a
pendulum. In Case 2 a point particle of mass 𝑀 is
attached to the center of the same stick.
Calculate the periods of oscillation. The moment
J
of inertia of a rod about its end is 𝐼 = 𝑀𝐿* ,
K
where 𝐿 is the length of the rod, and the moment
of inertia of a point mass is 𝐼 = 𝑀𝑟 * , where 𝑟 is
the distance between the mass and the pivot.