Physics Notes

https://www.sharelatex.com/blog/2012/06/12/mathtools-for-beautiful-math.html

5. Seeasaws

• How does a seesaw move?
  • Seesaw exhibits rotational, not translational, movement
  • Seesaw rotates around it’s C.O.M, it’s natural pivot
• Why does the seesaw need a pivot?
  • Prevent seesaw falling down due to gravity
  • Prevent unwanted rotations in different directions
  • If gravity was off wouldn’t need physical pivot
• Why does a lone seesaw rider plummet to the ground?
  • Seesaw is usually balanced, meaning it exhibits no torque due to gravity (obeys 1st law)
    • Because Centre of Gravity is above the pivot, as a result force is not at a lever arm from C.O.R and does not produce a torque about its pivot
    • Turns according to Newton’s 1st law of rotational motion; it’s a rigid object that is not wobbling and is not experiencing any external torques (twists)
    • Thus rotates at constant angular velocity
  • Lone rider causes seesaw to experience torque due to gravity and therefore angular acceleration
• How do rider’s weights and posistion affect the seesaw’s motion?
• Rider’s produce their own torques on the seesaw
  • If riders are balanced there is no gravitational torque on the riders or seesaw
    • the seesaw is rotationally intertial and moves according to 1st law
• If the riders are not balanced the seesaw undergoes angular acceleration according to newton’s second law

• Newton’s second law: Acceleration of seesaw = net torque acting on seesaw / rotational mass

• Why does the riders’ distance from the pivot affect the seesaw’s responsiveness?
  • The further riders are from the natural pivot the larger contribution to the overall rotational mass of the seesaw
    • Sit close to pivot = small R.M
    • Sit far from pivot = large R.M
• How do the seesaw riders affect one another?
  • Newtons Third law of Rotational Motion
    • As riders tip up and down they exert torques on one another according to newtons third law of rotational motion
    • If red rider exerts torque on purple rider, the purple rider has to exert an equal but oppositely directed torque on the red rider
  • Transfer work: As riders tip up and down they do work on each other
    • transfer energy with the help of the seesaw board
• Suppose you and a child half your weight lean out over the edge of a swimming pool at the same angle. If you both let go simultaneously, and tip into the water, which of you will hit the water first?
  • What is going to determine who hits the water first?
    • Angular acceleration. The person with the largest angular acceleration will undergo the largest angular velocity and tip over the fastest
  • What is the axis of rotation about which the swimmers rotate?
    • Not the natural pivot, C.O.M, but the feet
  • What is the resistance and cause of angular acceleration?
    • acceleration = net torque / rotational mass
    • cause of acceleration is net torque
    • resistance to acceleration is rotational mass
  • The adult is twice as tall and double the weight of the child
  • Answer: Child hits water first. Adult experiences double the torque, but mass is quadrupled. This means adult’s mass dominates. So, according Netwon’s 2nd law of acceleration, adult’s acceleration is half of child’s.

One Dimenisional Motion

Kinematic Formulas

• Solve missing value of object under constant acceleration

• Set of formulas related to the five following variables
  • \(\Delta x\) displacement
  • \(t\) time
    • t = amount of time it took the object to be displaced \(\Delta x\). It thus should be \(\Delta t\) but due to cluttering, is written as \(t\). \(t\) actually means \(\Delta t\).
  • \(v_o\) initial velocity
  • \(v\) = final velocity
  • \(a\) = constant acceleration
• If we know three of these 5 variables for an object under constant acceleration, we can use kinematic forumula to solve for missing value
  • \(v=v_0 +at\)
  • \(\Delta x=(\frac{v+v_0}{2})t\)
  • \(\Delta x= v_0t+ \frac{1}{2}at^2\)
  • \(v^2=v_0^2 + 2a\Delta x\)

Kinematic motion

• Distance travelled by flying objects
• constant acceleration = kinematic forumla

Problems without acceleration

• \(\Delta x = \frac{vf+v0}{2}t\)
• example:

A gritty chicken with admirable willpower is flying in a straight line with a speed of \(4.25 m/s\). A stiff breeze that lasts for \(1.5s\) causes the chicken to slow down with constant acceleration to a speed of \(2.75 m/s\) How far did the chicken fly during the stiff breeze?

• If both sides divided by time, Average velocity = the average of the velocities
• Final velocity of Chicken was \(vf = 2.75m\), initial velocity was \(v0 = 4.25m\)
• Time interval being considered was \(1.5 s\)
• \(\Delta x = \frac{vf+v0}{2}t\) = \(\Delta x = \frac{2.75 m/s+ 4.25 m/s}{2}(1.5 s) = 5.25m\)

Displacement, Velocity and time

• The posistion of a falling object accelerating due to gravity \(posistion = g * time^2/2\)

• Amount of time a dropped object takes
• Method 1: Coursera/Prem’s method
  • Re-arrange \(position = g * time^2/2\) to solve for time
  • divide by g.
    • \(position/ g = time^2/2\)
  • times by 2.
    • \(2*position/g = time^2\)
  • find square root.
    • \(\sqrt{2*position/g} = time\)
• Method 2: Khan academy Method
  • Set up co-ordinate system
    • y will be height. \(y=0m\) on the ground, y is positive above the ground.
    • Initial height = \(y0 = 13.7 m\)
    • Distance travelled = \(\Delta y = yf - y0 = 0m - 13.7m = -13.7 m\)
  • Write down initial velocity
    • \(v0 = 0 \frac{m}{s}\)
  • Write down acceleration due to gravity
    • \(a = -9.81\frac{m}{s^2}\)
  • There is a kinmatic Formula which relates vars \(\Delta y\), \(a\) and \(v_o\)
    • \(\Delta y = v_ot + \frac{1}{2}at^2\)
  • Re-arranged for time it is…
    • \(t = \sqrt\frac{2Delta y}{a}\)

• Example: Frustrated from a long day of physics problems, a student drops a physics book out of a window that’s 13.7m off the ground. How long did it take the book to hit the ground after being dropped?

# re-arrange pos = g*time^2/2
# time = sqrt(2*pos/g)
# pos = displacement (final pos - starting pos)
sqrt((2*13.7)/9.81)

## [1] 1.671248