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Chapter 6

Chapter 6. Motion In Two-Dimensional. Motion in Two Dimensions. Using + or – signs is not always sufficient to fully describe motion in more than one dimension. Vectors can be used to more fully describe motion. Still interested in displacement, velocity, and acceleration.

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Chapter 6

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  1. Chapter 6 Motion In Two-Dimensional

  2. Motion in Two Dimensions • Using + or – signs is not always sufficient to fully describe motion in more than one dimension. • Vectors can be used to more fully describe motion. • Still interested in displacement, velocity, and acceleration. • We can describe each of these in terms of vectors.

  3. Displacement • The position of an object is described by its position vector, . • The displacement of the object is defined as the change in its position.

  4. Velocity • The average velocity is the ratio of the displacement to the time interval for the displacement. • The instantaneous velocity is the limit of the average velocity as Δt approaches zero. • The direction of the instantaneous velocity is along a line that is tangent to the path of the particle and in the direction of motion.

  5. Acceleration • The average acceleration is defined as the rate at which the velocity changes. • The instantaneous acceleration is the limit of the average acceleration as Δt approaches zero.

  6. Unit Summary (SI) • Displacement • m (All displacement has the dimensions of length, L) • Average velocity and instantaneous velocity • m/s (All velocities have the dimensions of L/T) • Average acceleration and instantaneous acceleration • m/s2 (All accelerations have the dimensions of L/T2.

  7. Ways an Object Might Accelerate • The magnitude of the velocity (the speed) can change. • The direction of the velocity can change. • Even though the magnitude is constant • Both the magnitude and the direction can change.

  8. Projectile Motion

  9. Projectile Motion • An object may move in both the x and y directions simultaneously. • It moves in two dimensions • The form of two dimensional motion we will deal with is an important special case called projectile motion.

  10. Assumptions of Projectile Motion • We may ignore air friction. • We may ignore the rotation of the earth. • Disregard the curvature of the earth. • With these assumptions, an object in projectile motion will follow a parabolic path.

  11. Characteristics of Projectile Motion • The x- and y-directions of motion are completely independent of each other. • The x-direction is uniform motion: • ax = 0 • The y-direction is free fall: • ay = -g • The initial velocity can be broken down into its x- and y-components:

  12. Projectile Motion

  13. Some Details About the Rules • x-direction: the horizontal motion • ax = 0 • This is constant throughout the entire motion. • ∆x = voxt • This is the only operative equation in the x-direction since there is uniform velocity in that direction. • The horizontal range is dependent on the projection angle and the initial speed.

  14. More Details About the Rules • y-direction: the vertical motion • Vy = 0 m/s (at height) • Free fall problem • a = -g • Take the positive direction as upward. • Uniformly accelerated motion, so use the kinematic equations to solve for this motion.

  15. Velocity of the Projectile • The velocity of the projectile at any point of its motion is the vector sum of its x- and y-components at that point: • Remember to be careful about the angle’s quadrant.

  16. Projectile Motion Summary • Provided air resistance is negligible, the horizontal component of the velocity remains constant. • Since ax = 0 • The vertical component of the velocity vy is equal to the free fall acceleration –g • The acceleration at the top of the trajectory is not zero.

  17. Projectile Motion Summary (cont.) • The vertical component of the velocity vy and the displacement in the y-direction are identical to those of a freely falling body. • Projectile motion can be described as a superposition of two independent motions in the x- and y-directions. • The projection angle (θ) that the velocity vector makes with the x-axis is given by: Θ = tan-1vy/vx

  18. Problem-Solving Strategy • Select a coordinate system and sketch the path of the projectile • Include initial and final positions, velocities, and accelerations • Resolve the initial velocity into x- and y-components • Treat the horizontal and vertical motions independently

  19. Problem-Solving Strategy (cont.) • Follow the techniques for solving problems with constant velocity to analyze the horizontal motion of the projectile. • Follow the techniques for solving problems with constant acceleration to analyze the vertical motion of the projectile.

  20. Sample Problem A long jumper leaves the ground at an angle of 20.0o to the horizontal and at a speed of 11.0 m/s. (a) How far does he jump? (Assume that his motion is equivalent to that of a particle.) (b) What maximum height is reached?

  21. CircularMotion • Specific type of 2-dimensional motion. • Uniform circular motion: • The movement of an object or particle trajectory at a constant speed around a circle with a fixed radius.

  22. Displacement During Circular Motion • As an object moves around a circle its velocity vector does not change lengths but its direction does. • To get the object’s velocity we need the change in displacement.

  23. Centripetal Acceleration • An object traveling in a circle, even though it moves with a constant speed, will have an acceleration. • The centripetal acceleration is due to the change in the direction of the velocity.

  24. Centripetal Acceleration • Centripetal refers to “center-seeking.” • The direction of the velocity changes. • The acceleration is directed toward the center of the circle of motion.

  25. Centripetal Acceleration • The magnitude of the centripetal acceleration is given by: • This direction is toward the center of the circle. • So, whenever an object is moving in a circular motion it has centripetal acceleration. • Force causing this acceleration will be discussed later.

  26. Forces Causing Centripetal Acceleration • There are two things happening as an object moves along a circular path: • Object wants to move along a straight line. • Object wants to move along the circular path. Which one wins out?

  27. Forces Causing Centripetal Acceleration • Newton’s Second Law says that the centripetal acceleration is accompanied by a force. • FC = maC • FC stands for any force that keeps an object following a circular path. • Tension in a string • Gravity • Force of friction

  28. Centripetal Force • General equation: • If the force vanishes, the object will move in a straight line tangent to the circle of motion. • Centripetal force is a classification that includes forces acting toward a central point. • It is not a force in itself.

  29. Centripetal Force In each example identify the centripetal force.

  30. Sample Problem A 13 g rubber stopper is attached to a 0.93 m string. The stopper is swung in a horizontal circle, making one revolution in 1.18 s. Find the tension force exerted by the string on the stopper.

  31. THE END Chapter 6 Motion In Two-Dimensional

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