Three Chapter 5

• Before we defined inertial mass (mass determined by using Newton's 2nd Law (F = ma)).
• Gravitational mass - mass determined by using the force due to gravity on an object.
  • Basically, you use an equal arm balance and standard masses to measure gravitational mass.

• Mass remains constant no matter where you go. If you take this equal arm balance to the moon along with the unknown mass, you will measure the same mass for the object. This is because g appears on both sides of the last equation. Even though the acceleration due to gravity changes to 1.6 m/s² on the moon, it changes by the same factor on both sides of the equation, and therefore you measure the same mass.
• Principle of equivalence - Inertial mass equals gravitational mass.
  • This principle is the reason we can use the more general term "mass" instead of using inertial mass or gravitational mass.

• Galileo proposed that all objects fall at the same rate.
• Robert Boyle proved that Galileo was correct by using a vacuum pump. He dropped a lead ball and a feather in a vacuum and found them to fall at the same rate.
• Reasons for this observation are.
  • F = ma tells us we need a larger force to accelerate a larger mass at a desired acceleration.
  • F_g = mg tells us that a larger mass has a larger force due to gravity on it.
  • If we have two masses, mass 1 having a mass two times that of mass 2, then we can see that mass 1 (by F = ma) would require twice the force as mass 2 to accelerate at -9.81 m/s². On the other hand, since the mass of mass 1 is twice that of mass 2, mass 1 (by F_g = mg) has twice the gravitational force on it. The overall effect is that the two would fall at the same rate of -9.81 m/s² when undergoing free fall in vacuum.

· Mathematically:

• We can say that a = g is independent of mass since mass canceled out.
• The acceleration due to gravity on a planet is -1.92 N/kg. How long will it take for an object to fall 25m?

• When air is present, free fall is resisted by air friction.

• The net force causing free fall in air is therefore a vector sum of the force due to gravity and the air resistance.

• As an object falls, it speeds up, and so Air resistance (F_a) increases which reduces F_net. You should be able to see this in the light of the above equation and proportionality statement.
  • Eventually, once the speed is high enough, F_a = -F_g and F_net = 0.
  • Once F_net = 0 the object no longer accelerates, where as initially it did accelerate.
• Terminal velocity - The maximum velocity an object reaches when falling freely in a resistive medium.
• Resistive medium - A fluid which resists the free flow of an object through it.
• Terminal velocity varies from one object to the next.
• Let us consider a styrofoam ball and a billiard ball of the same size. Because they have the same size, they will have the same air resistance at the same speed.

• In the last equation, k is a proportionality constant. This last equation tells us the condition under which an object reaches terminal velocity (Acceleration equals zero.). From this equation you can see that if mass is large, then since k and g are constants, velocity (v) must be large in order for terminal velocity to be reached. This tells us that large mass objects will fall faster through resistive mediums like air than small mass objects.
• For our styrofoam ball and billiard ball, since the mass of the billiard ball is larger than the mass of the styrofoam ball, it will fall faster in air.
• Summarizing:
  • Differences in terminal velocity depend upon
    • differences in air resistance (due to different sizes and shapes of objects).
    • differences in mass.
• If we look at a velocity time graph for free fall in air we see:

• In section A of the graph, air resistance is almost zero, and therefore net force is approximately equal to the gravitational force. Therefore the acceleration is nearly constant which produces an almost straight diagonal line.
• In section B air resistance increases resulting in decreasing net force. Therefore the acceleration decreases over time. Remember that acceleration is the slope of the tangent to the line, and so as time goes by, this tangent must become less steep.
• Section C finds the air resistance equal to the gravitational force. This means the net force is zero and uniform motion is therefore present.
• Terminal velocity for us is about 200 km/h.
• For a parachute, when it opens:
  • Fa > Fg and Fnet is upwards. Therefore the rate of falling decreases to somewhere between 3 and 4.3 m/s when terminal velocity is reached.
  • Given enough time, Fa = Fg and terminal velocity is reached.

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November 5, 2013