Lecture 7: Forces and Motion in TWO Dimensions Read: Ch. 4.3 – 4.6 PHYS 218: General Physics Summer 2013 Lecture 7: Forces and Motion in TWO Dimensions Read: Ch. 4.3 – 4.6 Point them to this page for a description of course procedures.

From Yesterday: Friction (1D) Parabolic Motion (2D)

Friction N Fpull Ffriction Kinetic: |Ffriction | =µk |N| Static: Fgrav |Ffriction| = |F| for |Ffriction | < µs |N| Static: What happens if you put something heavy on top of the object that you are trying to move? Gravitational force goes up. Normal force goes up. You can push even harder and the object won’t move. Eventually it starts to move but only at a larger force. So, the frictional force depends on the forces in the vertical direction. Give some simple example. It’s easy to slide a piece of plywood across the floor right? What about sliding a piece of plywood with a person sitting on it? This is why N appears in the equation. Empirically, its found that the frictional force is roughly proportional to the normal force (incidentally, this is why I hate mixing this in the Newton’s Laws. Newton’s Laws are real physical laws that are true for nearly everything that the students ever come in contact. This is just a rule of thumb concerning the way somethings slide). If a=0 |Ffriction| = |F| Otherwise Fnet = |Ffriction| - |F|

Special Cases of 2D motion 2D Equilibrium: Special Cases of 2D motion

Example: Given what are and such that the system remains static? 20o This goes with a demo y x

So the actual free body diagram should look like, Do example!!! F1 = 150N This goes with a demo y x

two-dimensional (2D) equilibrium y x N Ffriction Fgrav q Example: What is the angle q when the block will first begin to slide?

At equilibrium (Newton’s 1st Law),

Finally, gives For 0.10 radians = 5.7 degrees

Things flying through the air under the force of gravity. (Yesterday) Projectile Motion Things flying through the air under the force of gravity. Kinematics

Chapter 4 Similar concepts to Chapter 3 (analysis of motion & Newton’s Laws) extended to 2-dimensions (2D). y x 1D All forces and motion in a single direction Force or motion in 2 directions 2D

Equation for motion with constant force/acceleration: Kinematic Equations: Equation for motion with constant force/acceleration: v(t) = v0 + a t x(t) = x0 + v0t + ½ a t2 v(t)2 =v02 + 2a(Δx)

What is the yellow ball’s horizontal position versus time? Quiz 1 What is the yellow ball’s horizontal position versus time? y x x position (m) x position (m) A B Time (sec) Time (sec) x position (m) x position (m) C D Time (sec) Time (sec)

Quiz 2 Does this graph represent the vertical position for the red ball or the yellow ball? y x y position (m) Time (sec) A. Red B. Yellow C. Both Red and Yellow

The point is x and y are completely decoupled here The point is x and y are completely decoupled here. The x-direction follows constant velocity, regardless of the fact that the ball is falling. Similarly, the object moves in y-direction just as it would had it been simply dropped.

Quiz 3 An object has a velocity of 1 m/s in the x-direction and a 1 m/s in the y-direction. What is the object’s speed in the x-direction? 1 m/s m/s 2 m/s None of the above y x v vy = 1 m/s vx = 1 m/s

Quiz 4 An object has a velocity of 1 m/s in the x-direction and a 1 m/s in the y-direction. What is the object’s speed? 1 m/s m/s 2 m/s None of the above y x v vy = 1 m/s vx = 1 m/s

q How to solve a 2D problem: First draw picture Then break into x and y components Write down what you know and want v vy = v sin  Choose equation Solve vx = v cos 

How to solve a 2D problem: A canon is fired at 35o above the horizontal at a muzzle velocity of 250m/s. How far does the cannonball fly? Do on board!!! q

Quiz 5 An object is fired from a cannon with an initial speed of 100 m/s at an angle of 30o. After 3 seconds has it hit the ground? Yes No q = 30o

Shoot the Monkey You fire the cannon. The monkey sees the flash and lets go to avoid the cannon ball. What angle should you set to hit the monkey? A. <q B. q C. >q. q

Why? In the absence of gravity, the monkey would hang there and the ball would follow the straight trajectory. Adding gravity to this causes both the monkey and the ball to accelerate at exactly the same rate. q

More violent physics You want to shoot the cannon as far as possible (to avoid the other guy’s guns). What q should you use? q

Trade-off between loft and forward velocity q = 90 deg q = 0 deg Lots of loft No forward velocity Lots of forward velocity no loft

Use independence of x and y x-component is easy. No forces means constant velocity q xlands

Use independence of x and y t is found by figuring out how long it would take the initial vy to to -vy. q xlands

Putting it together q xlands

Easiest way to find maximum q xlands

Quiz 6 (Prelude to circular motion) You peg the speedometer at constant speed 70 mph and whizz around a curved on-ramp. Are you accelerating? A. Yes B. No

Quiz (Prelude to circular motion) vx ~ 0 vy ~ v vx ~ -v vy ~ 0 y x You peg the speedometer at constant speed 70 mph and whizz around a curved on-ramp. Are you accelerating? A. Yes B. No

Quiz 6 (Prelude to circular motion) vx ~ 0 vy ~ v vx ~ -v vy ~ 0 A change in direction produces an acceleration. That’s why you feel a force when you drive around a curve. y x You peg the speedometer at constant speed 70 mph and whizz around a curved on-ramp. Are you accelerating? A. Yes B. No

For Next Time Read Chapter 5.1-5.3