PHYS 218 Chapter 3

« previous | Monday, January 31, 2011 | next »

Motion in 2 Dimensions

• Velocity is tangent to an object's path.
• Acceleration always points toward the concave side of an object's path

Any change in an object's path is the result of acceleration (you can feel acceleration). This means that even with a constant speed, velocity might change.

Acceleration parallel to velocity only speeds up object (magnitude)
Acceleration perpendicular to velocity maintains speed but changes direction

Circular Motion

A particle moving around a circle (stays same distance from center)

At two instances, the angles between the change around the radius and the change in velocity are equivalent

${\frac {\Delta {\overrightarrow {v}}}{|{\overrightarrow {v}}_{1}|}}={\frac {\Delta {\overrightarrow {r}}}{|{\overrightarrow {r}}_{1}|}}$

Always true in circular motion:

• $a_{\perp }={\frac {|{\overrightarrow {v}}|^{2}}{R}}$ (always to center)
• $a_{\parallel {\mbox{ to }}{\overrightarrow {v}}}={\frac {d|{\overrightarrow {v}}|}{dt}}$

If velocity is not constant around the entire circle, the resultant acceleration vector always points inward, but not necessarily toward center.

If the velocity remains constant around the entire circle, the parallel acceleration is 0, and the perpendicular velocity is a constant pointing to the center

Example

Imagine a perfectly circular race track with an even bank around the edge. A car is traveling at a constant speed around the track. The only (resultant) acceleration acting on the car is pointing toward the center.

Uniform Circular Motion

${\begin{cases}a_{\parallel }=0\\a_{\perp }={\frac {|{\overrightarrow {v}}|^{2}}{r}}\end{cases}}$

Period: time needed to make a full circle.

$|{\overrightarrow {v}}|={\frac {2\pi r}{t}}$

$a_{\perp }={\frac {({\frac {2\pi r}{t}})^{2}}{r}}={\frac {4\pi ^{2}r}{t^{2}}}$

Example

Pilot endurance test:

• r = 6m + 2m = 8m
• frequency = 2 rev/sec ∴ t = 1/2 sec
• $a_{\perp }={\frac {4\pi ^{2}(8)}{\frac {1}{4}}}$

Note: look through previous exams for sample problems

Relative Velocities

Position, Velocity, and Acceleration of an object as seen in reference to another position, velocity and acceleration in respect to each other

Draw two coordinate systems for this problem:

• x_p/A = x_B/A + x_p/B
• v_p/A = v_B/A + v_p/B
• a_p/A = a_B/A + a_p/B

Example

• Plane travels North at 300 km/h (with respect to the air)
• Wind is in Northeast direction at 50 km/h (with respect to ground)
• Find velocity of plane with respect to ground:

Given:

• v_air/ground = (50 km/h, 45°)
• v_{plane/air</sub = (300 km/h, 0°)}

$v_{\mbox{plane/ground}}=50\cos(45^{\circ }){\hat {x}}+50\sin(45^{\circ }){\hat {y}}+300{\hat {y}}$