Two Dimensional Motion

Two Dimensional Motion One Dimensional Motion – Describes Motion in a straight line Two Dimensional Motion – Describes, Motion in a Curved Path (Circular Motion and Projectile Motion) Two Dimensional Force Vectors (Equilibrant Vector and Inclined Planes)

Equilibrium If the Net Force acting on the object is zero FNET = 0 a = 0 The object is either stationary (v = 0) or traveling with a constant velocity (v = constant)

Equilibrant The Equilibrant (-R) is the single force that when added to a group of forces, produces Equilibrium. The Equilibrant (-R) is equal and opposite to the Resultant Force (R).

To Find the Equilibrant Find the Resultant (R) of the group of force vectors. R = A + B + C Draw the Equilibrant (-R) with the same magnitude in the opposite direction as the Resultant (R)

Inclined Planes An Inclined Plane is a flat surface with one end elevated higher than the other The Weight is a force that always acts downward Since the Weight Vector (mg) acts straight downward, the Weight Vector splits into 2 perpendicular components

Weight Components on an Inclined Plane One component of the weight acts parallel to the surface of the incline and down the incline (mg||) The other component of the weight acts perpendicular to the surface of the incline and into the incline (mg|)

Weight Components on an Inclined Plane mg|| - component of the weight acting parallel to the incline mg|– component of the weight acting perpendicular to the incline

Weight Components on an Inclined Plane Since the components are perpendicular to each other they form a right triangle and we can find the components with the sine and cosine functions mg|| = mg sin(q) mg| = mg cos(q)

Weight Components on an Inclined Plane Note: That as q increases mgsin(q) increases and mgcos(q) decreases These components of the weight are forces and can be used on a FBD with Newton’s Laws (Net Force Equation)

Projectile Motion A Projectile is anything thrown, shot or dropped into the air Projectile Motion is a combination of two independent motions Horizontal Motion (x-motion) Vertical Motion (y-motion)

Perpendicular Vectors Vectors that are Perpendicular to each other are INDEPENDENT of each other. Ex. Boat traveling perpendicular to the current. Ex. Projectile thrown in the air.

Perpendicular Vectors and Projectile Motion Since Projectile Motion is a combination of two perpendicular motions (vectors), each motion is independent of each other! Gravity (y-direction) does not affect the Horizontal Speed (vX) vX has no affect on vY

Projectile Motion There are no forces affecting the Horizontal Motion (aX = 0). We use only vX = dX/t for the x-motion The Vertical Motion is accelerated by gravity (aY = g). We use the Kinematic Equations for y-motion

Projectile Motion Problems There are three types of Projectile Motion Problems Objects shot straight upwards (vX = 0) Objects shot horizontally (half of a parabola) Objects shot at an angle (whole parabola)

Projectile Motion Terminology dX – Range – horizontal distance traveled dY – How High the projectile traveled vertically tH – half time – time the projectile travels upward or downward (Bottom to Top or Top to Bottom) tT – Total time – the total time the projectile is in the air

Uniform Circular Motion Uniform Circular Motion (UCM) describes an object traveling in a circular path at a constant speed Does an object in U.C.M. accelerate? YES!! Because there is a change in Direction!!

Finding the Acceleration for Uniform Circular Motion We cannot use a = (vf – vi)/t for uniform circular motion because the change in velocity is a change in direction (not a change in speed)! We need to derive another acceleration equation that uses vector diagrams to quantify this change in direction

Centripetal Acceleration ac = centripetal acceleration ac = (circular speed)2 radius of the circle ac = vc2/r S.I. Unit = m/s2

Centripetal Acceleration Acceleration that causes a change in direction. An object in circular motion has a centripetal acceleration. Centripetal (means “center-seeking”) Acceleration always acts towards the center of the circle.

Circular Speed Circular speed (vc) is the speed an object travels in a circle with. vc = distance = circumference time time vc = 2pr/t

Centripetal Force Centripetal (Fc) Force is the Force that causes a change in direction We can use Newton’s 2nd Law to calculate the Centripetal Force (Fc) Fc = mac The Centripetal Force (Fc) is a NET FORCE

Vertical Circle In a Horizontal Circle, gravity acts the same at every point in the circle In a Vertical Circle, gravity speeds up the object on the way down and slows down an object on the way up We need to use the Net Force Equation at different points in the circle to describe the motion

Vertical Circle The Centripetal Force is a NET FORCE! An object in a Vertical Circle usually has a Maximum Velocity at the Bottom of the circle and a Minimum at the Top A Vertical Circle has a minimum speed that must be maintained in order to stay in a circular path (Critical Velocity)

Rotary Motion Rotation or Rotary Motion is motion around a central axis (not a central point). Axis is the line that goes through the object around which the object rotates

Torque and Rotary Motion Translational (Linear) Motion is changed by a FORCE! Rotary Motion is changed by a TORQUE!

Torque Torque (t) causes rotation (analogous to Force in Linear Motion) Torque = Force x Lever Arm = t = Fl Lever Arm (l) is the perpendicular distance from the force to the axis line

Torque Directions for Torque (Vector) Clockwise (-) Counter Clockwise (+) Unit = Newton-meter (Nm) Torques can balance just like Forces (Clockwise and Counter-Clockwise)

Frames of Reference Frame of Reference or Reference Point are the objects used to determine what motion has taken place There are two types of frames Inertial Frames (Non-accelerated) Non-inertial Frames (Accelerated)

Frames of Reference Inertial Reference Points are not being accelerated (v=0 or v=constant). Newton’s laws are easily observable Non-inertial Reference Points are accelerated frames of reference and Newton’s Laws appear to break down in these reference frames

Centrifugal Force and Non-inertial Reference Frames Centrifugal Force – is the force that seems to push an object to the outside of the circular or curved path The Centrifugal Force is fictitious because the object is in an accelerated frame of reference and can be explained with Newton’s First Law