8.01SC | Fall 2016 | Undergraduate

Classical Mechanics

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Definition of Average Force:

The time average of a force \(\vec{F}(t)\) between times \(t_1\) and \(t_2\) is defined as:

\[\displaystyle \vec{F}_{ave} = \frac{\int_{t_1}^{t_2}\vec{F}(t)dt}{t_2-t_1}\]

A ball of mass \(\displaystyle m\) is released from rest from a height \(\displaystyle h_ i\) above a horizontal surface. It hits the surface and bounces off vertically to reach a maximum height \(\displaystyle h_ f\). The ball is in contact with the table for a time \(\displaystyle T\). Calculate \(\displaystyle N_{ave}\), the magnitude of the time average normal force exerted by the table on the ball. Express your answer in terms of \(\displaystyle m\), \(\displaystyle g\), \(\displaystyle T\), \(\displaystyle h_ i\), and \(\displaystyle h_ f\).

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Center of mass of a system of extended objects.

You can encounter a situation where a system is constituted of more than one extended object. To find the center of mass of a system of this kind:

1. Replace each of the extended objects by a point particle.

2. The mass of each point particle is the mass of the corresponding extended object.

3. The position of each point particle is the position of the center of mass of the corresponding object.

4. Finally, compute the position of the center of mass of the system from the positions and masses of these point particles.

This procedure is summarized in the figure below: Calculate the center of mass (CM) of the system formed by two square boxes, the lower one with a mass 2 times the mass of the upper box.


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Motion of the center of mass and the point particle approximation.

1. The center of mass of a rigid object moves like a point particle (explain the point particle approximation done so far)

2. An external impulse changes the velocity of the CM

3. No external force implies constant Vcm

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Problem Set 5 contains the following problems:

  1. Stopping a Bullet
  2. Acrobat and Clown
  3. Compressive Strength of Bones
  4. Center of Mass of a Rod
  5. Two Particles Colliding

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