1. Energy, Springs, Power, Examples
Physics 1D03 - Lecture 22

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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?
the heavy ball
the light ball
they will go up the same amount
Physics 1D03 - Lecture 22

9. 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

10. 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 —  kW
Physics 1D03 - Lecture 22

11. 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

12. 10 min rest
Physics 1D03 - Lecture 22

13. Oscillatory Motion - Chapter 14
Kinematics of Simple Harmonic Motion
Mass on a spring
Energy
Knight sections
Physics 1D03 - Lecture 22

14. 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

15. Spring and mass M M 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
For an ideal spring, the force is proportional to displacement. For this particular force behaviour, the oscillation is simple harmonic motion. x
F = -kx
Physics 1D03 - Lecture 22

16. 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

17. SHM: x(t) A = amplitude t f = phase constant w = angular frequency
A is the maximum value of x (x ranges from +A to -A).
f gives the initial position at t=0: x(0) = A cosf .
w 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

18. The period T of the motion is the time needed to repeat the cycle:
The quantity (w t + f) is called the phase, and is measured in radians. The cosine function traces out one complete cycle when the phase changes by 2p radians. The phase is not a physical angle!
The period T of the motion is the time needed to repeat the cycle:
units: radians/second or s-1
Physics 1D03 - Lecture 22

19. What is the oscillation period of a FM radio station with a signal at
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

20. Velocity and Acceleration
a(t) =- w 2 x(t)
Physics 1D03 - Lecture 22

21. Position, Velocity and Acceleration
x(t) t
v(t) t
a(t) t
Physics 1D03 - Lecture 22

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

23. 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.
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

24. 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). M v0 x
Result: x(t) = (5 cm) cos[π t – π/2]
Physics 1D03 - Lecture 22

25. 10 min rest
Physics 1D03 - Lecture 22

26. When do we have Simple Harmonic Motion ?
A system exhibits SHM is we find that acceleration is
directly proportional to displacement:
a(t) = - w 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

27. Mass and Spring M Newton’s 2nd Law: F = -kx so x
This is a 2nd order differential equation for the function x(t). Recall that for SHM, a = -w 2 x : the above is identical except for the proportionality constant. Hence, a spring/mass is a SHO. Hence, we must have:
or:
Physics 1D03 - Lecture 22

28. Recall: Velocity and Acceleration
We could use x=Asin(ωt+Φ) and obtain the same result
Physics 1D03 - Lecture 22

29. 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

30. 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.
find the period of its motion
determine the maximum speed of the block
determine the maximum acceleration of the block
Physics 1D03 - Lecture 22

31. 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

32. 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)?
Yes No
Yes, but only if it bounces to the same height each time
Physics 1D03 - Lecture 22