2. One just isn’t enough!

3. Four Major Types of Two Dimensional Motion 1. Projectile Motion 2. Circular Motion 3. Rotational Motion 4. Periodic Motion

5. Projectile motion problems are best solved by treating horizontal and vertical motion separately because perpendicular vectors are independent *IMPORTANT* Gravity only affects vertical motion. There are two general types of projectile motion situations. 1. o oo object launched horizontally 2. o oo object launched at an angle The path of a projectile is called the trajectory

7. Object Launched Horizontally v x = initial horizontal velocity R x = horizontal range h = initial height above ground t = total time in the air IMPORTANT FACTS There is no horizontal acceleration. There is no initial vertical velocity. The horizontal velocity is constant. Time is the same for both vertical and horizontal. horizontal Rx = vxt vertical h = 0.5gt2

9. Horizontal projectile examples

10. Object Launched at an Angle v = initial velocity  = launch angle h = maximum height t = total time in air R x = horizontal range IMPORTANT FACTS The horizontal velocity is constant. It rises and falls in equal time intervals. It reaches maximum height in half the total time. Gravity only effects the vertical motion. horizontal vx = v cos Rx = vxt vertical vy = v sin h = vyt/4 t = 2vy/g

13. Angled projectile examples:

14. Learn more about projectile motion at these links: link1link1, link2, link3, link4, link5, link6 link2link3link4link5link6 link1link2link3link4link5link6 View projectile motion simulations at: link1link1, link2, link3, link4, link5, link6 link2link3link4link5link6 link1link2link3link4link5link6

15. constant initial velocity versus how the horizontal range changes with angle; plot “range vs angle”constant initial velocity versus how the horizontal range changes with angle; plot “range vs angle” constant initial velocity versus how total time in air changes with angle; plot “total time vs angle”constant initial velocity versus how total time in air changes with angle; plot “total time vs angle” constant initial velocity versus how maximum height changes with angle; plot “height vs angle”constant initial velocity versus how maximum height changes with angle; plot “height vs angle” constant angle versus how the horizontal range changes with initial velocity; plot “range vs velocity”constant angle versus how the horizontal range changes with initial velocity; plot “range vs velocity” constant angle versus how the total time in the air changes with initial velocity; plot “time vs velocity”constant angle versus how the total time in the air changes with initial velocity; plot “time vs velocity” constant angle versus how the maximum height changes with initial velocity; plot “height vs velocity”constant angle versus how the maximum height changes with initial velocity; plot “height vs velocity” Suggested Constructivist Activities Students use simulations to complete data tables and make graphs of the following situations:

16. object moves in circular path about an External point (“orbits”)

17. According to Newton’s First Law of Motion, objects move in a straight line unless a force makes them turn. An external force is necessary to make an object follow a circular path. This force is called a CENTRIPETAL CENTRIPETAL (“center (“center seeking”) seeking”) FORCE. Since every every unbalanced force causes an object to accelerate in the direction of that force (Newton’s (Newton’s Second Law), Law), a centripetal force causes a CENTRIPETAL ACCELERATION. ACCELERATION. This acceleration results from a change in direction, and does not imply a change in speed, although speed may also change.

18. Centripetal force and acceleration may be caused by: gravity - planets orbiting the sungravity - planets orbiting the sun friction - car rounding a curvefriction - car rounding a curve a rope or cord - swinging a mass on a stringa rope or cord - swinging a mass on a string r m In all cases, a mass m moves in a circular path of radius r with a linear speed v. The time to make one complete revolution is known as the period, T. v The speed v is the circumference divided by the period. v = 2r/T If the force that causes it to turn is removed, it will travel in a straight line tangent to the circular path at the point the force was removed.

19. The formula for centripetal acceleration is: ac = v2/r and centripetal force is: Fc = mac = mv2/r m = mass in kg v = linear velocity in m/s F c = centripetal force in N r = radius of curvature in m a c = centripetal acceleration in m/s 2

20. For vertical circles: V is not constant through the circular path -V accelerates downward -V decelerates upward -F c is minimum on top and maximum on bottom because F c = mV 2 R -At the top, the force in the string and force of weight both act toward the center (downward) -At the bottom, the force in the string acts toward the center and the weight acts downward in an opposite direction.

21. At the top where V = V min : Fc is equal to F T and F W (F T + F W ) since both act toward the center. Therefore: F C(top) = F T + F W = mV min 2 r FTFT FWFW

22. At the bottom where V = V max, F c is F T minus F w (since they now act in opposite directions). Therefore, F c(bottom) = F T – F w = mV max 2 F c(bottom) = F T – F w = mV max 2 r Since V max is at the bottom, then F c is greater at the bottom; therefore F T is greatest at the bottom and greater than the F c at the bottom by an amount equal to the F w.

23. You can calculate the critical velocity, or minimum velocity required to maintain a circular path (such as that of a satellite about the earth or a roller coaster at the top of a loop) with V min = rg from: mV min 2 = F w (since F T = 0 at minimum r velocity b/f falling out r velocity b/f falling out of circular path) of circular path) mV min 2 = mg mV min 2 = mg r rmg = mV min 2 rmg = mV min 2 rg = V min 2 rg = V min 2 rg = V min rg = V min

24. Learn more about circular motion at these links: link1link1, link2, link3, link4, link5 link2link3link4link5 link1link2link3link4link5 View circular motion simulations at: link1link1, link2, link3, link4 link2link3link4 link1link2link3link4

25. object moves in circular path about an Internal point or axis (“rotates” or “spins”)

26. The amount that an object rotates is its angular displacement. angular displacement, , is given in degrees, radians, or rotations. 1 rotation = 360 deg = 2 radians The time rate change of an object’s angular displacement i ii is its angular velocity. angular velocity, , is given in deg/s, rad/s, rpm, etc...

27. Radians- an angle of one radian is the angle that, when placed with vertex at the center of a circle, substends on the circumference an arc equal in length to the radius of the circle: when s = r,  = 1 radian Since C = 2pr, 360 o = 2  radians and 1 radian = 57.3 o (a pure number; no unit)

28. Angular velocity is a vector. The “right-hand rule” describes the direction of angular velocity. The direction of  is the direction of the thumb of the right hand when the fingers curl in the rotational direction.

29. The time rate change of an object’s angular velocity is its angular acceleration. Angular acceleration, , is given in deg/s2, rad/s2, rpm/s, etc... Formulas for rotational motion follow an exact parallel with linear motion formulas. The only difference is a change in variables and a slight change in their meanings.

30. Constant LINEAR v f = v i + at d = v av t v av = (v f + v i )/2 d = v i t + 0.5at 2 v f 2 = v i 2 + 2ad ROTATIONAL  f  =  i +  t  =  av t v av =  (  f  +  i )/2  =  i t  +  0.5  t 2  f 2  =  i 2  +  2 

31. Forces that stop, start, or change the direction of rotation are called torque. Torque is equal to the product of the force applied and the lever arm The lever arm ( d, below) is the perpendicular distance from the axis of rotation to a line along which the force acts (the “F”, below. It was extended with dashed line downward to make it long enough to draw in the lever arm perpendicularly)

32. Rotational Inertia (I) is the resistance of a rotating object to changes in its angular velocity. We can convert the equation F = ma into one to be used for rotary motion: T =  (Where T = torque) Rotational inertia depends not only on the mass of the rotating object but also on the distribution of the mass:

33. PERODIC MOTION any motion in which the path of the object repeats itself in equal time intervals The simple pendulum is a great example of this type of motion.

34. When an object in periodic motion has one position in the motion where it is in equilibrium and is subject to a restoring force at all other positions that varies linearly with displacement from the equilibrium position, the object is in simple harmonic motion. Examples include a pendulum, an object moving up and down on a spring, and a guitar string vibrating. the restoring force, so called because it restores the object to the equilibrium position, may be gravity or the inherent stretch of a spring or guitar string. the restoring force, so called because it restores the object to the equilibrium position, may be gravity or the inherent stretch of a spring or guitar string. The period (T) is the time required to complete one full cycle of motion. The period (T) is the time required to complete one full cycle of motion. * The amplitude is the maximum distance the object moves from the equilibrium position.

35. The swing of a pendulum is simple harmonic motion: * the length of the string is l and the force of the suspended object is the weight which is resolved into 2 components. The F ll along the direction of the string and the The F ll along the direction of the string and the F l is at right angles to the direction of the string and always directed toward the equilibrium position. F l is at right angles to the direction of the string and always directed toward the equilibrium position. The F l is the restoring force for a pendulum. The F l is the restoring force for a pendulum.

36. The period, T, of a simple pendulum (time needed for one complete cycle) is approximated by the equation: where l is the length of the pendulum and g is the acceleration of gravity. The frequency of a pendulum is the number of complete cycles in one second. It is found by f = 1/T. The reciprocal of the period.

37. For pendulums, the period is independent of mass or material of the pendulum (neglecting air resistance), independent of amplitude (if the arc is small), and inversely proportional to the square root of the acceleration of gravity.

38. Learn more about pendulums and periodic motion at these links: link1link1, link2, link3, link4, link5 link2link3link4link5 link1link2link3link4link5 View pendulum simulations at: link1link1, link2, link3, link4, link5 link2link3link4link5 link1link2link3link4link5