Three Chapter 1

Centre of Mass

• Centre of Mass - The point where the entire mass of an object can be considered to act.
  • At geometric centre for regular shaped objects of uniform composition.
  • To locate the center of mass (cm)
    • Suspend an object by 2 different points
    • The center of mass is located at the intersections of the lines drawn vertically down from the suspension points.
  • For isolated systems, the centre of mass of the system will behave according to the laws of physics (Newton's three laws), regardless of how the parts of the system interact with each other.
• Consider:

• The ratio of the masses is

• The 6 m is divided in the inverse ratio. That is, if X is the distance from cm to the 6 kg
• object, then 2X is the distance from cm to the 3 kg object
• Therefore, 2X + X = 6 m or solving for X gives X = 2m (the distance from the 6 kg
• object to cm).
• For collisions of objects, the momentum of the centre of mass before interaction equals the momentum of the centre of mass after interaction.

Angular Momentum

• The momentum discussed so far is linear or translational momentum.
• Angular Momentum - The measure of the motion of an object that is rotating.
• An object can have both linear and angular momentum

• I = moment of inertia, w = angular velocity, and l = angular momentum.
• angular velocity - rotational rate
  • is measured in degrees per second or radians per second.
• radian - 2p radians = 360^o
  • a different unit for measuring angles
• Moment of inertia - a measure of an object's resistance to changes in rotational speed.
  • Resistance to angular acceleration
• For a system of objects:

• I = moment of inertia, r_i = the distance of the i th part of the system from the axis of rotation, and m_i = the mass of the i th part of the system.
• The formula for moment of inertia tells us that the closer the objects in the system are to the axis of rotation, the smaller the moment of inertia is.
  • The smaller the moment of inertia is, the less resistance to changes in angular momentum
• Just like linear momentum, angular momentum is conserved. That is to say, angular momentum equals a constant for a spinning object.
  • Therefore, if I decreases, (the ice skater in a spin pulls arms in), then w (angular velocity) must increase (The skater spins more quickly).
• Conservation of angular momentum has many applications.
  • used in
    • governors on motors
    • flywheels on large machinery
    • stability of gyroscopes - used for navigation and stabilizers for space craft.

Stopping Free Fall

• When a bungee jumper jumps, velocity must change from some initial velocity when the cord becomes tight, to 0 m/s.
  • There is a change in momentum present

DP = mDV

• This quantity DP equals a constant for any one jumper and length of cord
• If the cord does not stretch, then the time for the change in velocity (momentum) is very small (DT is approximately equal to zero).
• Impulse applied to this situation tells us

FDT = mDV = a constant

• The smaller DT, the greater the force needed to stop the jumper. Therefore if the cord doesn't stretch (DT is approximately equal to zero) then the force (F) is very large. On the otherhand, if the cord stretches then DT increases and the force (F) is smaller.
• The more the cord stretches, the more the jumpers body can withstand the forces.
• For a bungee cord, the force changes as the cord stretches according to the formula:

F = kx

Movember 30, 2013