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

Physics 1D03 - Lecture 22

Energy, Springs, Power, Examples


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.


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.


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 ?


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.


Example 5v0

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?


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?


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


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


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 kWMetric horsepower — 0.735 kW

Electrical horsepower — 0.746 kW Boiler horsepower — 9.8095 kW


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


10 min rest


Oscillatory Motion -Chapter 14

• Kinematics of Simple Harmonic Motion• Mass on a spring• Energy

Knight sections 14.1-14.6


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

Oscillatory Motion


Spring and mass

MEquilibrium: no net force

M

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


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=0B) at x=AC) at x=-AD) both at x=A and x=-A


)cos( tAxSHM: x(t)

t

A

-A

T

A = amplitude

= phase constant

= angular frequency

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.


)360(or radians 2 if

)0()( so

)2cos( cos(0)

T

xTx

AAx

2 2

fT

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:

units: radians/second or s-1


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?


Velocity and Acceleration

2MAX

MAX

22

:Note

)cos()(

)sin()(

)cos( )(

Aa

Av

xtAdt

dvta

tAdt

dxtv

tAtx

a(t) 2 x(t)


Position, Velocity and Acceleration

x(t) t

v(t) t

a(t) t


Question:

Where during the motion is the velocity largest?

Where during the motion is acceleration largest?

When do these happen ?


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) determine the amplitudeb) determine the frequencyc) determine the periodd) its position at t=0 sec

e) calculate the velocity at any time, and the vmax

f) calculate the acceleration at any time, and amax

Example


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

x

v0

Result: x(t) = (5 cm) cos[π t – π/2]


10 min rest


When do we have Simple Harmonic Motion ?

a(t) = 2 x(t)

A system exhibits SHM is we find that acceleration is directly proportional to displacement:

SHM is also called ‘oscillatory’ motion. Its is called ‘harmonic’ because the sine and cosine function arecalled harmonic functions, and they are solutions to the abovedifferential equation – lets prove it !!!

SHM is ‘periodic’.


Mass and Spring

xmk

dtxd

a 2

2

M

x

F = -kxkxmaF Newton’s 2nd Law:

so

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.

Hence, we must have: mk2 m

kor:


Recall: Velocity and Acceleration

xtAdt

dvta

tAdt

dxtv

tAtx

22 )cos()(

)sin()(

)cos( )(

We could use x=Asin(ωt+Φ) and obtain the same result


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).

Example


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) find the period of its motion

b) determine the maximum speed of the block

c) determine the maximum acceleration of the block


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 ???


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) Yes

b) No

c) Yes, but only if it bounces to the same height each time

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

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