Force and linear motion relationships linear kinematics

1. Force and linear motion relationships(linear kinematics) Newton’s first law, states the resultant force acting on a body must be equal to zero, when the body is at rest or constant velocity. Newton’s second law relates the resultant force to the acceleration of the body’s mass. According to Newton’s third law, the force of action-reaction is equal in magnitude and opposite in direction to the force of reaction.

2. Contents Cartesian reference system Linear speed and velocity Linear acceleration Acceleration due to gravity Constant velocity – zero acceleration Variable velocity Centripetal force and radial acceleration

3. Cartesian reference system A reference axis (coordinate system) enables forces to be described graphically. A coordinate system in which the coordinates of a point are its distances from a set of perpendicular lines that intersect at an origin, such as two lines in a plane or three in space.

4. Coordinates Right hand coordinate system z - positive upwards A Position vector is given by: Magnitude and direction Velocity is a vector. Therefore, velocities add like vectors.

7. Cartesian reference system used in sport

8. A 2-D global reference system and a 2-D reference system that defines the local knee joint centre.

9. Example When you are given a coordinate system! P = 1000 m ? = 30° What are the x and y components of P Px = P cos 30° = 866 m Py = P sin 30° = 500 m Note: Given Px and Py, what is P? P2 = Px2 + Py2=8662 + 5002 = 1000 m

10. Motion It is convenient to describe motion in terms of space and time. This portion of mechanics is called kinematics. It is important to understand motion occurs in three-dimensions in the human body. There are three types of motion in biomechanics: translation, rotation and vibration (outside our field of study).

11. Motion descriptors Joint centres or segment centres of mass are frequently used to describe, or quantify human motion. These points need to be accurately identified in space and time.

12. Joint centre

13. Regression equations, estimating body segments weights and locations of centre of mass

14. Estimating segment weight using regression equations. For example. If an individual weighed 850 N then, Trunk weight = 0.532 x 850 – 6.93 = 445.27 N. This accounts for 53% of total body weight.

15. Estimating segment centre of mass location using regression equations If the length of an individual’s segment length is known; centre of mass location can be estimated. For example. If the length of an individual’s thigh is (hip to knee distance) is 36 cm and the centre of mass for that thigh is located 39.8% of that distance from the hip joint. CM location = 36 x 0.398 (% in decimal format) CM = 14.3 cm from the hip joint

16. Motion descriptors Movement involves the shift from one position to another. Motion can be described in terms of the shift size (displacement) or the rate at which it occurs (velocity and acceleration). Displacement is the “space” element of motion; it is defined by how far the object has moved from its start and the direction it has moved.

17. Displacement and distance Displacement is the straight line from one position to the next. Displacement is how far an object travelled and in which direction. Displacement is therefore a vector. Distance is a scaler because it only records how far an object travelled which is the actual length of the path taken.

18. Position and displacement

19. Displacement ?y = yf-yi ?x = = xf-xi ? y = 3m-1m = 2m ? = 2m-1m = 1m The high-jumper is displaced 2m vertically and 1m horizontally. What is the resultant of this displacement vector? What is the direction of the displacement?

20. Velocity and speed Velocity is a vector quantity defined as the time rate of change of position. Speed is scalar and describes magnitude i.e. how much speed an object has. Velocity = displacement / time V = positionf – positioni time at final position – time at initial position V = ? position / ? time Velocity is measured in m.s-1

21. Horizontal position plotted as a function of time

22. Using the position-time data below calculate the average velocity from frame 3-5

23. Position-Velocity-time graphs

24. Instantaneous velocity When time (t) becomes smaller and smaller, the calculated velocity is the average over a much briefer time interval. In this process of making t smaller it will eventually approach zero known as the limit. This is instantaneous velocity. Instantaneous velocity is the tangent to the position-time curve.

25. Instantaneous velocity

26. Instantaneous speed The speedometer of a car reveals information about the instantaneous speed of your car; that is, it shows your speed at a particular instant in time.

27. Graphical velocity curve based on the shape of a position-time profile

28. Relative velocity The state of motion can be described in terms of being at rest (not moving relative to the ground) or in terms of motion with a given speed.

29. Relative velocity

30. Linear acceleration Acceleration is the rate of change of velocity with respect to time. a = v2 - v1 t2 - t1 Note: Acceleration can be zero when velocity is constant, or it can negative or positive depending upon the direction of the acceleration. Acceleration can also be viewed as the second derivative of position, making it length /time2. Acceleration can be written as 5 m/s/s, m/s2 or m·s-2.

31. Average acceleration

32. Calculation of acceleration from a set of velocity-time data

33. Constant acceleration

34. Constant acceleration equations

35. Constant acceleration equations When the initial velocity is zero. vf = at d = 1/2 at2 vf2 = 2ad Can these equations be used when acceleration varies with time? Can they be used when acceleration is zero?

36. Kinematics of motion Positive Velocity and Positive Acceleration Observe that the object below moves in the positive direction with a changing velocity. An object which moves in the positive direction has a positive velocity. If the object is speeding up, then its acceleration vector is directed in the same direction as its motion (in this case, a positive acceleration). The “dots" shows that each consecutive dot is not the same distance apart (i.e., a changing velocity). The position-time graph shows that the slope is changing (meaning a changing velocity) and positive (meaning a positive velocity). The velocity-time graph shows a line with a positive (upward) slope (meaning that there is a positive acceleration); the line is located in the positive region of the graph (corresponding to a positive velocity). The acceleration-time graph shows a horizontal line in the positive region of the graph (meaning a positive acceleration).

37. Kinematics of motion Positive Velocity and Positive Acceleration

38. Kinematics of motion Positive Velocity and Negative Acceleration Observe that the object below moves in the positive direction with a changing velocity. An object which moves in the positive direction has a positive velocity. If the object is slowing down then its acceleration vector is directed in the opposite direction as its motion (in this case, a negative acceleration). The “dots" shows that each consecutive dot is not the same distance apart (i.e., a changing velocity). The position-time graph shows that the slope is changing (meaning a changing velocity) and positive (meaning a positive velocity). The velocity-time graph shows a line with a negative (downward) slope (meaning that there is a negative acceleration); the line is located in the positive region of the graph (corresponding to a positive velocity). The acceleration-time graph shows a horizontal line in the negative region of the graph (meaning a negative acceleration).

39. Kinematics of motion Positive Velocity and Negative Acceleration

40. Kinematics of motion Negative Velocity and Negative Acceleration Observe that the object below moves in the negative direction with a changing velocity. An object which moves in the negative direction has a negative velocity. If the object is speeding up then its acceleration vector is directed in the same direction as its motion (in this case, a negative acceleration). The “dots" shows that each consecutive dot is not the same distance apart (i.e., a changing velocity). The position-time graph shows that the slope is changing (meaning a changing velocity) and negative (meaning a negative velocity). The velocity-time graph shows a line with a negative (downward) slope (meaning that there is a negative acceleration); the line is located in the negative region of the graph (corresponding to a negative velocity). The acceleration-time graph shows a horizontal line in the negative region of the graph (meaning a negative acceleration).

41. Kinematics of motion Negative Velocity and Negative Acceleration

42. Kinematics of motion Negative Velocity and Positive Acceleration Observe that the object below moves in the negative direction with a changing velocity. An object which moves in the negative direction has a negative velocity. If the object is slowing down then its acceleration vector is directed in the opposite direction as its motion (in this case, a positive acceleration). The “dots" shows that each consecutive dot is not the same distance apart (i.e., a changing velocity). The position-time graph shows that the slope is changing (meaning a changing velocity) and negative (meaning a negative velocity). The velocity-time graph shows a line with a positive (upward) slope (meaning that there is a positive acceleration); the line is located in the negative region of the graph (corresponding to a negative velocity). The acceleration-time graph shows a horizontal line in the positive region of the graph (meaning a positive acceleration).

43. Kinematics of motion Negative Velocity and Positive Acceleration

44. Question A diver of mass 60 kg leaps of a 10 m tower. Calculate their velocity after 4 m. Solution: First find the time: d = 1/2 at2 T = 0.9 sec Then using a = ?v -9.81 = Vfinal - 0 ? t 0.9 Final velocity at 4 m = -8.86 ms-1.

45. Constant acceleration equations In two -dimensional motion having constant acceleration is equivalent to two independent motions in the x and y directions having constant acceleration ax and ay. This is most notable in projectile motion with the horizontal acceleration component (ax) (negating air resistance) is zero, while the vertical (ay) is a constant.

46. Instantaneous acceleration

47. Acceleration due to gravity Newton’s law of gravitation states: “All bodies attract one another with a force proportional to the product of their masses and is inversely proportional to the square of the distance between them.” Greater the mass or the closer the distance, the greater the force of attraction. Acceleration of gravity (g) on earth is -9.81 m/ s2 Gravity decreases with increasing altitude on earth.

48. Acceleration due to gravity-free falling objects An object thrown upward and one thrown downward will both experience the same acceleration as an object released at rest. Once they are in free-fall, all objects have an acceleration downward, equal to free-fall acceleration.

49. Radial and Tangential acceleration The velocity of a rotating body changes in direction and magnitude. The velocity vector is always tangent to the path. As the body moves along the curved path the direction of total acceleration vector a, changes from point to point. This vector can be resolved into two components: radial ar and tangential at component. Therefore a = ar + at . The tangential acceleration arises from the change in speed of the particle, and the projection of the acceleration along the direction of the velocity: at = v /t The radial acceleration is due to the change in direction of the velocity vector and is: ar = v2/r

50. Radial and Tangential acceleration

51. Question A ball tired to the end of a strong of .50 m in length swings in a vertical circle under the influence of gravity as seen below. When the string makes an angle of ? of 20° with the vertical, the ball speed is 1.5 ms-1. Find the magnitude of the radial component of acceleration at this instant.