Normal-Tangential coordinates Lecture IV The description of the Plane Curvilinear Motion by the normal-tangential (n-t) coordinates & the Polar Coordinates Normal-Tangential coordinates Polar coordinates

Plane Curvilinear Motion – Normal-Tangential (n-t) Coordinates Here, the curvilinear motions measurements are made along the tangent (t) and the normal (n) to the path. n-t coordinates are considered to move along the path with the particle. The positive direction of the normal (n) always points to the center of curvature of the path; while the positive direction of the tangent (t) is taken in the direction of particle advance (for convenience). et & en are the unit vectors in t-direction and n-direction, respectively.

(n-t) Coordinates - Velocity Note: r is the radius of curvature and db is the increment in the angle (in radians) Note: as mentioned before that the velocity vector v is always tangent to the path; thus, the velocity has only one component in the n-t coordinates, which is in the t-direction. This means that vn = 0. Its magnitude is: (after dt)

(n-t) Coordinates - Acceleration Note: et, in this case, has a non-zero derivative, since it changes its direction. Its magnitude remains constant at 1. (after dt) ? Note: the vector det , in the limit, has a magnitude equal to the length of the arc |et|db=db. The direction of det is given by en. Thus,

(n-t) Coordinates – Acceleration (Cont.) Notes: an always directed toward the center of curvature. at positive if the speed v is increasing and negative if v is decreasing. Its magnitude is: r = , thus an = 0

(n-t) Coordinates – Circular Motion For a circular path: r = r

n-t Coordinates Exercises

Exercise # 1 2/97: A particle moves in a circular path of 0.4 m radius. Calculate the magnitude a of the acceleration of the particle (a) if its speed is constant at 0.6 m/s and (b) if its speed is 0.6 m/s but is increasing at the rate of 1.2 m/s each second. .

Exercise # 2 2/101: The driver of the truck has an acceleration of 0.4g as the truck passes over the top A of the hump in the road at constant speed. The radius of curvature of the road at the top of the hump is 98 m, and the center of mass G of the driver (considered a particle) is 2 m above the road. Calculate the speed v of the truck.

Exercise # 3 2/110: Write the vector expression for the acceleration a of the mass center G of the simple pendulum in both n-t and x-y coordinates for the instant when q = 60° if q. = 2.00 rad/s and q.. = 2.45 rad/s2.

Exercise # 4 2/118: The design of a camshaft-drive system of a four cylinder automobile engine is shown. As the engine is revved up, the belt speed v changes uniformly from 3 m/s to 6 m/s over a two-second interval. Calculate the magnitudes of the accelerations of points P1 and P2 halfway through this time interval.

Exercise # 5 2/128: The pin P is constrained to move in the slotted guides which move at right angles to one another. At the instant represented, A has a velocity to the right of 0.2 m/s which is decreasing at the rate of 0.75 m/s each second. At the same time, B is moving down with a velocity of 0.15 m/s which is decreasing at the rate of 0.5 m/s each second. For this instant determine the radius of curvature r of the path followed by P.

Plane Curvilinear Motion – Polar Coordinates Here, the curvilinear motions measurements are made by the radial distance (r) from a fixed pole and by an angular measurement (q) to the radial line. The x-axis is used as a reference line for the measurement of q. er & eq are the unit vectors in r-direction and q-direction, respectively.

Polar Coordinates – Position & Velocity Note: from (b), der is in the positive q-direction and deq in the negative r-direction The position vector of the particle: The velocity is: ? (after dt) (after dt)

Polar Coordinates – Velocity (Cont.) Thus, the velocity is: Its magnitude is: Due to the rate at which the vector stretches Due to rotation of r

Polar Coordinates - Acceleration Rearranging, Centripetal acceleration Its magnitude is: Coriolis acceleration

Polar Coordinates – Circular Motion For a circular path: r = constant Note: The positive r-direction is in the negative n-direction, i.e. ar = - an

Polar Coordinates Exercises

Exercise # 6 2/131: The boom OAB pivots about point O, while section AB simultaneously extends from within section OA. Determine the velocity and acceleration of the center B of the pulley for the following conditions: q = 20°, q . = 5 deg/s, q .. = 2 deg/s2, l = 2 m, l . = 0.5 m/s, l .. = -1.2 m/s2. The quantities l . and l .. are the first and second time derivatives, respectively, of the length l of section AB.

Exercise # 7 2/133: The position of the slider P in the rotating slotted arm OA is controlled by a power screw as shown. At the instant represented, q . = 8 rad/s and q .. = -20 rad/s2. Also at this same instant, r = 200 mm, r. = -300 mm/s, and r.. = 0. For this instant determine the r- and q-components of the acceleration of P .

Exercise # 8 2/142: At the bottom of a loop in the vertical (r-q) plane at an altitude of 400 m, the airplane P has a horizontal velocity of 600 km/h and no horizontal acceleration. The radius of curvature of the loop is 1200 m. For the radar tracking at O, determine the recorded values of r.. and q.. for this instant.