Curvilinear Motion: Normal and Tangential Components

Size: px

Start display at page:

Download "Curvilinear Motion: Normal and Tangential Components"

Julius Atkinson
5 years ago
Views:

1 Curvilinear Motion: Normal and Tangential Components

2 Coordinate System Provided the path of the particle is known, we can establish a set of n and t coordinates having a fixed origin, which is coincident with the particle at the instant considered.

3 The positive tangent axis acts in the direction of motion and the positive normal axis is directed toward the path's center of curvature.

4 Velocity The particle's velocity is always tangent to the path. The magnitude of velocity is found from the time derivative of the path function. v = ds dt

5 Tangential Acceleration The tangential component of acceleration is the result of the time rate of change in the magnitude of velocity. a t = dv dt

6 This component acts in the positive s direction if the particle's speed is increasing or in the opposite direction if the speed is decreasing.

7 Normal Acceleration The normal component of acceleration is the result of the time rate of change in the direction of the velocity. a n = v2 ρ

8 This component is always directed toward the center of curvature of the path 2 2 n t a a a

9 Radius of Curvature If the path is expressed as y = f(x), the radius of curvature ρ at any point on the path is determined from the equation 1 dy dx 2 d y 2 3/2 dx 2

10 Example 1 When the skier reaches point A along the parabolic path, he has a speed of 6 m/s which is increasing at 2 m/s 2. Determine the direction of his velocity and the direction and magnitude of his acceleration at this instant. Neglect the size of the skier in the calculation.

11 Example 2 A race car C travels around the horizontal circular track that has a radius of 300 ft. If the car increases its speed at a constant rate of 7 ft/s 2, starting from rest, determine the time needed for it to reach an acceleration of 8 ft/s 2. What is its speed at this instant?

12 Example 3 The boxes travel along the industrial conveyor. If a box starts from rest at A and increases its speed such that at = (0.2t) m/s 2, where t is in seconds, determine the magnitude of its acceleration when it arrives at point B.

13 Activity The boat is traveling along the circular path with a speed of v = (0.0625t 2 ) m/s, where t is in seconds. Determine the magnitude of its acceleration when t = 10 s.

14 Curvilinear Motion r, θ

15 Coordinate System If the path of motion is expressed in polar coordinates, then the velocity and acceleration components can be related to the time derivatives of r and θ.

16 Velocity v v r dr dt d r dt

17 Acceleration a r 2 2 d r dt 2 r d dt d 2 dr d a r 2 dt 2 dt dt

18 Example 1 The rod OA rotates in the horizontal plane such that θ = (t 3 ) rad. At the same time, the collar B is sliding outward along OA so that r = (100 t 2 ) mm. If in both cases t is in seconds, determine the velocity and acceleration of the collar when t = 1 s.

19 Example 2 Due to the rotation of the forked rod, the ball travels around the slotted path, a portion of which is in the shape of a cardioid, r = 0.5(1 - cos 8) ft, where 8 is in radians. If the ball's velocity is v = 4 ft/s and its acceleration is a = 30 ft/s2 at the instant... 8 = 180, determine the angular velocity 8 and angular acceleration 8 of the fork.

20 Activity A particle moves along a circular path of radius 300 mm. If its angular velocity is θ = (2t 2 ) rad/s, where t is in seconds, determine the magnitude of the particle's acceleration when t = 2 s.

21 Absolute Dependent Motion Analysis of Two Particles

22 In some types of problems the motion of one particle will depend on the corresponding motion of another particle. This dependency commonly occurs if the particles, here represented by blocks, are interconnected by inextensible cords which are wrapped around pulleys.

23 For example, the movement of block A downward along the inclined plane will cause a corresponding movement of block B up the other incline.

24 Procedure for Analysis Position-Coordinate Equation 1) Establish each position coordinate with an origin located at a fixed point or datum. 2) Using geometry or trigonometry, relate the position coordinates to the total length of the cord. exclude the segments that do not change length as the particles move-such as arc segments wrapped over pulleys.

25 Time Derivatives. 3) Two successive time derivatives of the positioncoordinate equations yield the required velocity and acceleration equations which relate the motions of the particles.

26 Example 1 Determine the speed of block A if block B has an upward speed of 6 ft/s.

27 Example 2 A man at A is hoisting a safe S by walking to the right with a constant velocity V A = 0.5 m/s. Determine the velocity and acceleration of the safe when it reaches the elevation of 10 m. The rope is 30 m long and passes over a small pulley at D.

28 Example 3 Determine the speed of block B if the end of the cord at A is pulled down with a speed of 2 m/s.

29 Activity Determine the speed of A if B has an upward speed of 6 ft/s.

Addis Ababa University Addis Ababa Institute of Technology School Of Mechanical and Industrial Engineering Extension Division` Assignment 1

Addis Ababa University Addis Ababa Institute of Technology School Of Mechanical and Industrial Engineering Extension Division` Assignment 1 Assignment 1 1. Vehicle B is stopped at a traffic light, as shown in the figure. At the instant that the light turns green, vehicle B starts to accelerate at 0.9144m/s 2. At this time vehicle A is 91.44m