Understanding Gyroscopes and Angular Momentum
Explore the concept of gyroscopes, angular momentum, and gyroscopic effects in vehicles like ships and airplanes. Learn about mass moment of inertia, angular velocity, and the precession of gyroscopes in detail. Dive into the physics behind the change of angular momentum and solve a problem involving a ship in motion.
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Presentation Transcript
OUTLINE 1. Concept of angular momentum (L). 2. Rate of change of angular momentum. 3. Gyroscopic Couple. 4. Gyroscopic Couple effect in a ship. 5. Gyroscopic effect in a aero plane.
Angular momentum (L) Angular momentum (L): If a circular disc rotating about an axis passing through center perpendicular to circular plane. Mass moment of inertia o f rotor (I): Mass moment of inertia about axis of rotation of rotor is I. For circular disc, Izz = M = Mass of rotor. of disc and M R2 2 Axis of rotation (Spin) y R = Radius of rotor. A A If radius of gyration of rotor is k, then mass x z Moment of inertia (Izz) = m k2 Unit : Kg - m2 Angular momentum about axis of rotation (L) B B Front view Side view L = Mass moment of inertia * Angular velocity L = I is angular velocity of rotor. Direction of angular momentum is same as Sense of rotation is anticlockwise direction. direction of angular velocity. Axis of rotation about z-axis. Unit : kg m2 / sec Angular velocity is vector quantity.
Angular Velocity representation Representation of angular velocity in vector form: Representation of angular velocity in vector form: Sense of rotation : Anticlockwise direction. Sense of rotation : Clockwise direction. Axis of rotation : + z axis Axis of rotation : - z axis Magnitude of angular speed : (rad/s) Magnitude of angular speed : (rad/s) + z - z + z - z L L
Axis of precession y y p A x z B p = axis of precision. Axis of precession is rotation of rotor about y-axis.
Change of angular momentum (L) i Time t = t + dt sec Time t = t sec x I d y p z' k I z d ? ? = ?? ? z ? ? + ?? = ?? ??? ? ? + ??? ? ? L t + dt L t z is axis of spin at time t = t + dt z is axis of spin at time t = t sec = I Sin d i + Cos d k I k ? ?? = p ? = p dt For small time dt, d is small, Sin d = d Cos d = 1 ??????????? ?????? = ?? p = I d ? p dt i dt = I
PROBLEM: Suppose a ship is moving in sea. At any instant of time it make a left turn with radius of curvature R. Y TOP VIEW R C X t = 0 s L L = Angular momentum. = I I = Mass moment of inertia about axis of spin. Rotor is rotating in anticlockwise direction. ROTOR
PROBLEM: Suppose a ship is moving in sea. At any instant of time it make a left turn with radius of curvature R. Y TOP VIEW p = angular velocity of precession d dt = p C X d Axis of Spin Lf d Lf Li Li time = dt L = Angular momentum. = I I = Mass moment of inertia about axis of spin.
PROBLEM: Suppose a ship is moving in sea. At any instant of time it make a left turn with radius of curvature R. Y TOP VIEW p = angular velocity of precession d dt = p Vector for of p. p = pk Initial angular momentum at time t =0 C X Li d Lf d Li L = Angular momentum. = I I = Mass moment of inertia about axis of spin.
Y Torque= ( I p )j Y X X Y X = Spin axis Lf Y = Active Gyroscopic couple d X Li Li = I i Lf = I ( cos(d ) i + Sin(d )j ) Lf - Li = I ( cos(d ) i + Sin(d )j ) I i Lf - Li = I ( i + (d )j ) I I = I d j Lf - Li = I d dt dt
Numerical 1. The turbine rotor of a ship has a mass of 2.2 tonnes and rotates at 180 rpm clockwise when viewed from the aft. Radius of gyration k = 320 mm, find (i) The ship turns at a radius of 250 m with a speed of 25 km/h. Find gyroscopic couple. Gyroscopic couple = I * * p . Mass moment of inertia (I) = m*k2 ( k = radius of gyration) I = 2200 kg * (0.32)2 m2 I = 225.28 kg m2 Angular velocity of spin ( ) = (2*3.14* N)/60 = (2*3.14* 180)/60 = 18.85 rad/sec. Angular velocity of precision ( p) = velocity of ship (v) / radius (R) = (25*1000)/ (250* 3600) = 0.03 rad/sec. Gyroscopic couple = I * * p = 225.28 * 18.85 * 0.03 C = 127.4 N-m
