Question

Tutorial: Angular Momentum and Torque III. Angular Momentum The angular momentum of a point particle is defined by: L = ixp. Here, is a vector that points from the point of rotation (or point around which the angular momentum is calculated) to the location of the particle and p = mb is the linear momentum of the particle. A Your little brother Joey is playing with his toy airplane. The airplane is tied to a string and its motor makes it go around a horizontal circle at constant speed. Your goal is to calculate the magnitude and direction of the angular momentum of the airplane (mass m, speed v, length of the string is a). You will do this in several steps: i. Draw the vector for one particular position of the airplane. ii. Assuming that the image is taken directly from above. What is the direction of the angular momentum of the airplane? Up word ill. What is the angle between the linear momentum of the plane and i'? Draw it in the figure. iv. What is the magnitude of the angular momentum of the airplane? Important note: Angular momentum can always be calculated for a moving object, even if the object is not tied to anything or rotating around a particular point or axis. This will be important when solving problems involving "circular collisions", for example, a bullet fired into the rim of a wheel making it spin. You will now practice calculating the angular momentum of a particle moving in a certain direction around specific points axes. P B. A particle of mass m is moving at a constant spoed to the right. At a certain point in time, the particle is directly below an axis perpendicular to d the page going through point P as shown. What is the magnitude and direction of the angular momentum of the particle around the axis going through point P7 Follow the same steps you did in the previous question. a C. At a later time, the particle has moved some distance to the right as shown below. You will again calculate the magnitude and direction of the angular momentum of the particle around the axis going through point P by following several steps. i. Draw the vector and use the picture to figure out its magnitude. ii. Identify the angle between the linear momentum of the particle and f. Remember that the angle is between these two vectors when they have the same origin. iii. What is the direction of the angular momentum of the particle? iv. What is the magnitude of the angular momentum of the particle?

Answer 1

at position of the airplane is drawn below.

direction of angular momentum of the airplane is given by cross product of position vector and momentum vector of airplane i. e. ř x B momentum is given by p = mở Here m is the mass of the air plane and is the velocity of airplane. This direction is given by right hand thumb rule. If fingers of right hand are curled from

then thumb point in the direction of angular momentum. Direction of angular momentum will be upward as shown by green colour.

If top view is taken as below momentum vector position Vector ? , angular momentum vector (pespeneliceby to plane of paper) (top view) The angle between linear momentum of the airplane and

90° This is shown with blue colour. momentum vector position vector 909 Xa angular te momentum vector (pespeneliceday to plane of paper) (top view)

iv) magnitude of angular momentum of the airplane \Ž=\PxP L=rpsin 0 L=rmvsin 0 Substitute r= a and A = 90° L = amvsin 90° L=mva

LI * - - - - ----> E E 11 momentum vector will be in the velocity vector. | 0--------------- l to Draw vector 'd' B.)

Direction of angular momentum will be out of the page, perpendicular to plane of paper outward as shown by green colour. Ozgo° ña Now The angle between linear momentum of the particle and position vector r is 0 = 90° magnitude of angular momentum of the particle is El=\ L=rpsin 0 L=rmvsin e

L=(d) x (mv) (sin 90°) L=mvd T is drawn below. 30°C m 1 1> from triangle ABC

r= 2d The angle between linear momentum of the particle and can be calculated (make same origin)

Direction of m O=30 direction of a 0 = 30° iii) Direction of angular momentum of the particle will be out of the page, perpendicular to plane of paper outward as shown by green colour.

30° nec 1 el m 1> magnitude of angular momentum of the particle is Ž[=]; xp | L=rpsin e L=rmysin e L=(2d) x (mv) * (sin30°) L = (2d)x (mv) x (1) L = mvd