1020

• Run Multiple Choice Generator at h:\shares\physics
• Run Core Problems by giving the program qzz as your 'course'--enter this when asked for your 3-digit course number.   Do the Physics Quiz 1018.
• If you miss a problem be sure to click on Comment and explain what you do and do not understand about the question and the given solution

University Physics students note:  instructions_for_program_on_basic_calculus

Graph PE vs. position for pendulum.

If PE increases as you move away from the equilibrium position then the equilbrium is stable.  Think of a pendulum at equilibrium.  If you push it to the right or left, the net force acting on the pendulum tends to move it back toward the equilibrium position.  So the net force is opposing your attempt to move it away from equilibrium, and PE is increasing.

If PE decreases as you move away from the equilibrium position then the equilbrium is unstable.  A meter stick standing on its end will lose PE as you move it away from the equilibrium position.

Systems tend to fall into 'low points' of PE vs. position graphs, and away from 'high points'.

Sines and Cosines

Angles are always measured from the positive x axis.

Draw a unit circle with the origin at the center.

For any angle theta find the point where the radial line intersects the circle.

sin(theta) is the y coordinate of that point

cos(theta) is the x coordinate of that point

You should be able to visually estimate within +-.1 the sine or cosine of any given angle.

You should be able to estimate within +-5 degrees the angle corresponding to any point on a unit circle.

You should then be able to use your calculator to check your visual estimates of the sine and the cosine.

You should finally be able to check the plausibility of your sine and cosine estimates by checking whether sin^2(theta) + cos^2(theta) is close to 1.  For any angle, sin^2(theta) + cos^2(theta) = 1 by the Pythagorean Theorem.

Note the following behaviors of the sine and cosine:

As we move a little ways from 90 degrees or from 270 degrees the sine changes very little, the cosine changes almost in proportion to how far we move.

As we move a little ways from 0 degrees or from 180 degrees the cosine changes very little, the sine changes almost in proportion to how far we move.

Applications of Vectors

• Pendulum

For small angles:

• Ty is close to T because sin(theta) is close to 1
• Very little vertical acceleration so Ty is close to mg (vertical equilibrum, nearly)
• So T is close to mg.
• Therefore Tx is close to m g cos(theta).

From a picture of the situation we easily figure out that cos(theta) = -x / L.

• Therefore Tx = m g * ( -x / L).

This is the net force on a free pendulum (small-angle approximation).

Thus

• Fnet = m g ( -x / L) or
• Fnet = -(m g / L) * x.

This is of the form

• Fnet = - k x

where k is a constant.

 

• Forces on Incline
• Pulling at non-horizontal angle on cart
• Projectiles with Nonzero Initial Vertical Velocity

Impulse-Momentum