Lecture 12: Non-Conservative Forces - Resistive Forces - Air Drag - Terminal Velocity
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1. Resistive and Drag Forces:
Resistive forces have a viscous term that is linear in velocity; it is temperature sensitive and reflects the stickiness of the medium. In addition they have a pressure term that is proportional to the speed squared and the fluid's density. Resistive forces are always in the direction opposite the velocity. The resistive force grows as the speed increases. Therefore, a falling object in air (or in a liquid) will reach a terminal velocity.
2. Two Regimes and the Critical Velocity:
The drag, or resistive force has two terms; the viscous and pressure terms. They are equal in magnitude at a "critical" speed. At speeds much smaller than this, the terminal speed for spherical objects (all with the same density) increases as the radius squared of the objects. At speeds much larger than the critical speed, the pressure term dominates and the terminal velocity increases as the square root of the radius.
3. Measurements with Steel Balls in Syrup:
Small ball bearings are dropped in Karo Corn Syrup. Professor Lewin explains why the terminal velocity of the ball bearings will vary with the radius squared. He conducts measurements to validate this.
4. Reaching Terminal Velocity in the Blink of an Eye:
When the ball bearings are dropped in the syrup, their speeds at first increase. It is shown that they reach their terminal velocity very fast.
5. Air Drag and the Pressure Term:
The air drag on almost all objects that fall in air from a considerable height (raindrops or sky divers) is dominated by the pressure term. Thus the terminal speed increases with the square root of the radius of spheres with given density. If you want to calculate the time it takes to reach this terminal speed, you have to include both the v and v-squared terms. Lewin's graduate student, Dave Pooley, solved the equation numerically, and a measurement is made with a balloon (filled with air) that is dropped from a height of about 3 meter.
6. Numerical Calculations of Air Drag Examples:
A pebble, dropped from a height of 475 meters (the Empire State Building), reaches a terminal speed of about 75 miles per hour in 5-6 seconds. Professor Lewin also discusses the contribution of air drag to the quantitative experiments done earlier in the course with falling apples.
7. Resistive Forces and Trajectories:
Air drag will result in an asymmetric trajectory for an object thrown up in the air. The resistive force is about the same for a tennis ball as for a styrofoam ball of the same radius, but the resistive force has a much more dramatic effect on the lighter ball's trajectory.
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