Chapter 3 Nonlinear Motion Scalar quantity ---- ------ a quantity that has magnitude but not...
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Transcript of Chapter 3 Nonlinear Motion Scalar quantity ---- ------ a quantity that has magnitude but not...
Chapter 3Chapter 3
Nonlinear MotionNonlinear Motion
• Scalar quantity ----• ------ a quantity that has
magnitude but not direction
• Vector quantity -• ------ a quantity that has both magnitude and direction
• Vector - • -------an arrow drawn to scale used
to represent a vector quantity
These represent equivalent vectors:
• Vector quantity - a quantity that has both magnitude and direction
• Vector - an arrow drawn to scale used to represent a vector quantity
• Scalar quantity - a quantity that has magnitude but not direction
ExamplesExamples
• Speed………..• Velocity……....• Acceleration..• Time………….• Distance……..• Force…………
scalarvectorvectorscalarscalarvector
Addition of VectorsAddition of Vectors• The sum of two or more vectors is called their
resultant.
• To find the resultant of two vectors that are at angles to each other, we use the tip-to-tail method.
Projectile MotionProjectile Motion• A projectile is any object that is projected by some means and continues in
motion by it own inertia.
• The velocity of a projectile has a horizontal and vertical component. Each component acts independently of the other.
• For the vertical motion the acceleration is 9.8m/s2 downward.
• For the horizontal motion there is no acceleration.
• Projectile Drawing.
Projectile MotionProjectile Motion
• The shape of a projectiles path is a parabola.
• The same range is obtained from two different projection angles that add up to 90°.
• Maximum range for a projectile is achieved with a projection angle of 45°.
• In the presence of air resistance, the trajectory of a high-speed projectile falls short of a parabolic path.
• A projectile fired horizontally will hit the ground at same time as an object dropped from rest if they are released at the same height.
• Demo: Ball projector and dropper
Example QuestionsExample Questions• You are driving along in an open car and throw
a ball straight up into the air. Neglect air resistance.
(a) Where does the ball land relative to the car?Answer: In the car.
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Example: Projectile Motion• An object may move in both the x and y directions
simultaneously (i.e. in two dimensions)• The form of two dimensional motion we will deal with is called
projectile motion
• We may: » ignore air friction
» ignore the rotation of the earth
• With these assumptions, an object in projectile motion will follow a parabolic path
Notes on Projectile Motion:
• once released, only gravity pulls on the object, just like in up-and-down motion
• since gravity pulls on the object downwards:
vertical acceleration downwards NO acceleration in horizontal direction
Projectile Motion
Rules of Projectile Motion
• Introduce coordinate frame: y is up• The x- and y-components of motion can be treated
independently• Velocities (incl. initial velocity) can be broken down into
its x- and y-components• The x-direction is uniform motion
ax = 0• The y-direction is free fall
|ay|= g
Some Details About the Rules
• x-direction – ax = 0– – x = vxot
• This is the only operative equation in the x-direction since there is uniform velocity in that direction
constantvcosvv xooxo
More Details About the Rules
• y-direction– – take the positive direction as upward– then: free fall problem
• only then: ay = -g (in general, |ay|= g)
– uniformly accelerated motion, so the motion equations all hold
ooyo sinvv
Velocity of the Projectile
• The velocity of the projectile at any point of its motion is the vector sum of its x and y components at that point
x
y12y
2x v
vtanandvvv
(b) While the ball is still in the air you step on the accelerator. Where does the ball land relative to the car?
Answer: Behind the car.
(c) What if you stepped on the brake instead?Answer: In front of the car.
• Demo: Cart and ball launcher
Example QuestionsExample Questions• You drop a ball from the window of a school bus
moving a 10 miles/hour. Neglect air resistance.
(a) Where does the ball land relative to your hand?Answer: Directly below your hand.
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(b) What is the shape of the path made by the ball seen by someone outside the bus?
Answer: A parabola.
Fast-Moving Projectiles SatellitesFast-Moving Projectiles Satellites
• An earth satellite is simply a projectile that falls around the earth rather that into it.
• Satellites are not free from gravity.
• The speed of the satellite must be great enough to ensure that its falling distance matches the earth's curvature.
• It takes 1 second for an object in free fall to fall 4.9 meters.
• Earth's surface 'drops' a vertical distance of 4.9 meters for every 8000 meters along the Earth's surface.
• This means that a satellite near the Earth’s surface must travel at 8000 meters/second!
• …or 18,000 miles per hour.
• For example: The space shuttle orbits the Earth once every 90 minutes.
Circular MotionCircular Motion• Linear speed - the distance moved per unit
time. Also called simply speed.
• Rotational speed - the number of rotations or revolutions per unit time.
• Rotational speed is often measured in revolutions per minute (RPM).
• The linear speed is directly proportional to both rotational speed and radial distance.
v = r
Example QuestionExample Question• Two ladybugs are sitting on a phonograph
record that rotates at 33 1/3 RPM.
(a) Which ladybug has a great linear speed?Answer: The one on the outside edge.
(b) Which ladybug has a great rotational speed?Answer: Both have the same
rotational speed.
*
You sit on a rotating platform halfway between the rotating axis and the outer edge.
You have a rotational speed of 20 RPM and a tangential speed of 2 m/s
What will be the rotational speed of your friend who sit at the outer edge?
Answer: 4 m/s
What will be his rotational speed?
Answer: 20 RPM
• See this question on page 50.
*
End of Chapter 3