Monday, October 5, 1998 Chapter 5: Springs Chapter 6: Linear Momentum Conservation of Momentum Impulse

A 600 kg elevator starts from rest and is pulled upward by a motor with a constant acceleration of 2 m/s 2 for 3 seconds. What is the average power output of the motor during this time period? F net = ma = (600 kg)(2 m/s 2 ) F net = 1200 N = F motor - F g = F motor - mg 1200 N = F motor - (600 kg)(10 m/s 2 ) = F motor N F motor = 6000 N N = 7200 N Let’s first figure out the force delivered by the motor...

A 600 kg elevator starts from rest and is pulled upward by a motor with a constant acceleration of 2 m/s 2 for 3 seconds. What is the average power output of the motor during this time period? Now we need to determine the work done by the motor... W = F  s But we don’t know  s, so…. s = s 0 +v 0 t +0.5at 2 = (2 m/s 2 )(3 s) 2 = 9 m W = (7200 N)(9 m) = J

In addition to the gravity, there are other mechanisms to store POTENTIAL ENERGY. One of them is... Sir Robert Hooke unlocked the secret of the spring... A spring resting in its natural state, with a length l exerts no horizontal force on anything!

However, if we compress or stretch the spring by some amount x, then the spring is observed to exert a Force in the opposite direction. Hook discovered this force could be modeled by the mathematical expression F = - kx Notice that this force operates along a linear line! l x

Which means that if we looked at the plot of Force versus compression/stretching x... Force x Slope of this line is - k, where k is the spring constant.

Force x If we look at the work done by an applied force which compresses the spring through a distance (- x 1 )... -x 1 F1F1 Work done BY the external force ON the spring. This energy is stored in the spring...

Potential Energy of a spring is So, for spring problems, we have a new TOTAL MECHANICAL ENERGY given by And it is THIS quantity which will be conserved absent other, outside forces.

Momentum & Collisions The linear momentum of an object of mass m moving with velocity v is defined as the product of the mass and the velocity:

Notice that momentum is a vector quantity, which means that it must be specified with both a magnitude and direction. Also notice that the direction of the momentum vector is necessarily parallel to the velocity vector.

OR The units suggest a relationship between force and momentum.

What happens when we apply a force to an object? It accelerates. Its velocity changes. The force imparts momentum. Its momentum changes.

By how much will the momentum change? That depends upon the length of time over which the force is applied to the object.

Impulse Change in momentum The impulse of a force on an object equals the change in momentum of that object. Notice that impulse is a vector quantity as well!