Transcript Potential Energy - McMaster Physics & Astronomy Outreach

Energy, Springs, Power, Examples
Physics 1D03 - Lecture 22
Example 1
A child of mass 30kg starts on top of a water slide, 6m above the
ground. After sliding down to a position of 1m above the
ground, the slide curves up and end 1m above the lowest
position before the child leaves it and falls into a pool of water.
Determine the speed at which the child leaves the slide.
Physics 1D03 - Lecture 22
Example 2
A child of mass 30kg starts on top of a water slide, 6m above the
ground. After sliding down to a position of 1m above the
ground, the slide curves up and end 1m above the lowest
position before the child leaves it and falls into a pool of water if
the total length of the slide was 20m and there was a constant
frictional force of 10 N. Determine the speed at which the child
leaves the slide.
Physics 1D03 - Lecture 22
Example 3
• A mass of 5kg is placed on a vertical spring with a
spring constant k=500N/m. What is the maximum
compression of the spring ?
Physics 1D03 - Lecture 22
Example 4
You slide 20m down a frictionless hill with a slope of 30o
starting from rest. At the bottom you collide and stick
to another person (at rest) that has 90% of your mass
and move on a level frictionless surface.
a) Determine the final velocity of the system.
b) Determine the velocity if the slope had a coefficient
of kinetic friction of 0.1.
Physics 1D03 - Lecture 22
Example 5
v0
A block of mass m = 2.0 kg
slides at speed v0 = 3.0 m/s
along a frictionless table
towards a spring of stiffness k
= 450 N/m. How far will the
spring compress before the
block stops?
Physics 1D03 - Lecture 22
Example 6
In the figure below, block 2 (mass 1.0 kg) is at rest on a frictionless
surface and touching the end of an unstretched spring of spring
constant 200 N/m. The other end of the spring is fixed to a wall.
Block 1 (mass 2.0 kg) , traveling at speed v1 = 4.0 m/s ,
collides with block 2, and the two block stick together. When
the blocks momentarily stop, by what distance is the spring
compressed?
Physics 1D03 - Lecture 22
Concept Quiz
Two identical vertical springs are compressed by the
same amount, one with a heavy ball and one with a
light-weight ball. When released, which ball will
reach more height?
a) the heavy ball
b) the light ball
c) they will go up the same amount
Physics 1D03 - Lecture 22
Power
The time rate of doing work is called power.
If an external force is applied to an object, and if work
is done by this force in a time interval Δt, the average
power is defined as:
P=W/Δt
(unit: J/s = Watt, W)
For instantaneous power, we would use the
derivative:
P=dW/dt
And since W=F.s, dW/dt=Fds/dt=F.v, sometimes it is
useful to write:
P=F . v
Physics 1D03 - Lecture 22
Power output
• A 100W light bulb :100J/s
• A person can generate about ~ 350 J/s
• A car engine provides about 110,000 J/s
Many common appliances are rated using
horsepower (motors for example):
1hp~745 W
Mechanical horsepower — 0.745 kW
Metric horsepower — 0.735 kW
Electrical horsepower — 0.746 kW
Boiler horsepower — 9.8095 kW
Physics 1D03 - Lecture 22
Example
An elevator motor delivers a constant force of 2x105N
over a period of 10s as the elevator moves 20m.
What is the power ?
P=W/t
=Fs/t
=(2x105N)(20m)/(10s)
=4x105 W
Physics 1D03 - Lecture 22
10 min rest
Physics 1D03 - Lecture 22
Oscillatory Motion Chapter 14
• Kinematics of Simple Harmonic Motion
• Mass on a spring
• Energy
Knight sections 14.1-14.6
Physics 1D03 - Lecture 22
Oscillatory Motion
We have examined the kinematics of linear motion with
uniform acceleration. There are other simple types of
motion.
Many phenomena are repetitive or oscillatory.
Example: Block and spring, pendulum, vibrations
(musical instruments, molecules)
M
Physics 1D03 - Lecture 22
Spring and mass
M
Equilibrium: no net force
The spring force is always directed back
towards equilibrium. This leads to an
oscillation of the block about the
equilibrium position.
M
F = -kx
M
For an ideal spring, the force is
proportional to displacement. For this
particular force behaviour, the oscillation
is simple harmonic motion.
x
Physics 1D03 - Lecture 22
Quiz:
You displace a mass from x=0 to x=A and let it go from
rest. Where during the motion is acceleration largest?
A) at x=0
B) at x=A
C) at x=-A
D) both at x=A and x=-A
Physics 1D03 - Lecture 22
SHM:
x  A cos(t   )
x(t)
A
T
A = amplitude
t
 = phase constant
 = angular frequency
-A
A is the maximum value of x (x ranges from +A to -A).
 gives the initial position at t=0: x(0) = A cos .
 is related to the period T and the frequency f = 1/T
T (period) is the time for one complete cycle (seconds).
Frequency f (cycles per second or hertz, Hz) is the number of
complete cycles per unit time.
Physics 1D03 - Lecture 22
The quantity ( t + ) is called the phase, and is measured in radians.
The cosine function traces out one complete cycle when the phase
changes by 2 radians. The phase is not a physical angle!
The period T of the motion is the time needed to repeat the cycle:
x(0)  A cos  A cos(2   )
so x(T )  x(0)
if  T  2 radians (or 360)
2

 2f
T
units: radians/second or s-1
Physics 1D03 - Lecture 22
Example - frequency
What is the oscillation period of a FM radio station with a signal at
100MHz ?
Example - frequency
A mass oscillating in SHM starts at x=A and has a period of T.
At what time, as a fraction of T, does it first pass through x=A/2?
Physics 1D03 - Lecture 22
Velocity and Acceleration
x(t )  A cos(t   )
dx
v(t ) 
  A sin(t   )
dt
dv
2
2
a(t ) 
  A cos(t   )   x
dt
a(t)   2 x(t)
Not e: vMAX  A
aMAX  A 2
Physics 1D03 - Lecture 22
Position, Velocity and Acceleration
x(t)
t
v(t)
t
a(t)
t
Physics 1D03 - Lecture 22
Question:
Where during the motion is the velocity largest?
Where during the motion is acceleration largest?
When do these happen ?
Physics 1D03 - Lecture 22
Example
An object oscillates with SHM along the x-axis. Its displacement from the
origin varies with time according to the equation:
x(t)=(4.0m)cos(πt+π/4)
where t is in seconds and the angles in radians.
a)
b)
c)
d)
e)
f)
determine the amplitude
determine the frequency
determine the period
its position at t=0 sec
calculate the velocity at any time, and the vmax
calculate the acceleration at any time, and amax
Physics 1D03 - Lecture 22
Example
The block is at its equilibrium position and
is set in motion by hitting it (and giving it a
positive initial velocity vo) at time t = 0. Its
motion is SHM with amplitude 5 cm and
period 2 seconds. Write the function x(t).
Result:
v0
M
x
x(t) = (5 cm) cos[π t – π/2]
Physics 1D03 - Lecture 22
10 min rest
Physics 1D03 - Lecture 22
When do we have Simple Harmonic Motion ?
A system exhibits SHM is we find that acceleration is
directly proportional to displacement:
a(t) =   2 x(t)
SHM is also called ‘oscillatory’ motion.
Its is called ‘harmonic’ because the sine and cosine function are
called harmonic functions, and they are solutions to the above
differential equation – lets prove it !!!
SHM is ‘periodic’.
Physics 1D03 - Lecture 22
Mass and Spring
Newton’s 2nd Law:
F  ma   kx
F = -kx
M
so
d 2x
k
a 2  x
dt
m
x
This is a 2nd order differential equation for the function x(t).
Recall that for SHM, a   2 x : the above is identical except for the
proportionality constant. Hence, a spring/mass is a SHO.
k
Hence, we must have:  
m
2
or:

k
m
Physics 1D03 - Lecture 22
Recall: Velocity and Acceleration
x(t )  A cos(t   )
dx
v(t ) 
  A sin(t   )
dt
dv
2
2
a (t ) 
  A cos(t   )   x
dt
We could use x=Asin(ωt+Φ) and obtain the same result
Physics 1D03 - Lecture 22
Example
A 7.0 kg mass is hung from the bottom end of a vertical spring
fastened to the ceiling. The mass is set into vertical oscillations
with a period of 2.6 s.
Find the spring constant (aka force constant of the spring).
Physics 1D03 - Lecture 22
Example
A block with a mass of 200g is connected to a light spring with a
spring constant k=5.0 N/m and is free to oscillate on a
horizontal frictionless surface.
The block is displaced 5.0cm from equilibrium and released from
rest.
a)
b)
c)
find the period of its motion
determine the maximum speed of the block
determine the maximum acceleration of the block
Physics 1D03 - Lecture 22
Example
A 1.00 kg mass on a frictionless surface is attached to a horizontal
spring. The spring is initially stretched by 0.10 m and the
mass is released from rest. The mass moves, and after 0.50
s, the speed of the mass is zero.
What is the maximum speed of the mass ???
Physics 1D03 - Lecture 22
Concept Quiz
A ball is dropped and keeps bouncing back after hitting the floor.
Could this motion be represented by simple harmonic motion
equation, x=Asin(ωt+Φ)?
a)
b)
c)
Yes
No
Yes, but only if it bounces to the same height each time
Physics 1D03 - Lecture 22