* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Example2 - mrdsample
Mitsubishi AWC wikipedia , lookup
Routhian mechanics wikipedia , lookup
Specific impulse wikipedia , lookup
Sagnac effect wikipedia , lookup
Classical mechanics wikipedia , lookup
Old quantum theory wikipedia , lookup
Coriolis force wikipedia , lookup
Laplace–Runge–Lenz vector wikipedia , lookup
Tensor operator wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Modified Newtonian dynamics wikipedia , lookup
Center of mass wikipedia , lookup
Moment of inertia wikipedia , lookup
Mass versus weight wikipedia , lookup
Fictitious force wikipedia , lookup
Seismometer wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Newton's theorem of revolving orbits wikipedia , lookup
Photon polarization wikipedia , lookup
Rotational spectroscopy wikipedia , lookup
Equations of motion wikipedia , lookup
Relativistic mechanics wikipedia , lookup
Newton's laws of motion wikipedia , lookup
Jerk (physics) wikipedia , lookup
Angular momentum wikipedia , lookup
Hunting oscillation wikipedia , lookup
Angular momentum operator wikipedia , lookup
Classical central-force problem wikipedia , lookup
Relativistic angular momentum wikipedia , lookup
Angle measurement can be defined in degrees but also can be defined in radians. The angle θ in radians is defined: Angular Quantities Angular displacement: The average angular velocity is defined as the total angular displacement divided by time: The angular acceleration is the rate at which the angular velocity changes with time: Converting between angular and linear Every point on a rotating body has an angular velocity ω and a linear velocity v. If the angular velocity of a rotating object changes, it has a tangential acceleration: Even if the angular velocity is constant, each point on the object has a centripetal acceleration: Constant Angular Acceleration The equations of motion for constant angular acceleration are the same as those for linear motion, with the substitution of the angular quantities for the linear ones. Example1 A record player turntable is turned on. It is noted that the disk turns 0.75 revolutions as it goes from rest to 33.3 rpm, clockwise. (a) Determine the disk’s final angular velocity in radians per second. (b) Determine the disk’s angular acceleration. Example2 A cyclist is traveling at 7.0 m/s to the right when she applies the brakes and slows to a speed of 5.0 m/s. Each wheel of the bicycle has radius 0.33 m and completes 5.0 revolutions during this braking period. Determine the time that elapses in this period. Example3 A string is wrapped around the axle of a gyroscope – 30.0 cm of the string is in contact with the axle, which has diameter 2.20 mm. Starting at rest the string is pulled with a constant acceleration, which causes the gyroscope to start spinning. It takes 1.10 seconds to pull the string off of the axle and the gyroscope then spins for an additional 60.0 seconds before stopping. (a) Find the maximum angular speed of the gyroscope in rad/s. (b) Find the total number of revolutions the gyroscope will spin. Torque To cause a change in rotation, a net torque is required. Torque is the force applied at a perpendicular distance from the pivot of rotation. Torque is defined as: Calculate the net torque and direction about a pivot at the center of the 1.0m long uniform beam 25N 30o 45o 20o 10N 30N Rotational Inertia The resistance to change an object’s state of rotation is called rotational inertia. Various shapes and formulas for rotational inertia. Depends on mass distribution and location of axis of rotation. General form of equation is: Rotational Dynamics Just like a net force can cause an acceleration according to FNET = ma A net torque can cause a rotational acceleration where Example A 150kg merry-go-round (treat as a solid disk) of radius 1.50m is set in motion by pulling on a rope that is wrapped around the rim. What constant force must be exerted on the rope to give the merry-go-round a speed of 0.50revs/sec in 2.0s? Example2 A 5.0kg pulley with radius 0.60m and frictionless axle starts from rest and speeds up uniformly as a 3.0kg mass falls making a light rope unwind from the pulley. The mass falls for 4.0s. Treat pulley as disk. a) What is the linear acceleration of the mass? b) How far does mass fall in the time stated? + c) What is the angular acceleration of the pulley? d) What is the tension in the rope? Example3 A potters wheel has a radius of 0.50m and a moment of inertia of 12kgm2 is rotating freely at 50rev/min. The potter can stop the wheel in 6.0s by pressing a finger against the edge and exerting a radial inward force of 70N. Find the coefficient of friction between wheel and finger. Rotational Kinetic Energy Recall that the translational kinetic energy of a moving object is given by Total Kinetic Energy Example Find the angular speed of a solid sphere (R = 0.2m) at the bottom of the 4.0m long incline if it starts from rest at the top. 20o Work & Rotational Kinetic Energy The torque does work as it moves the wheel through an angle θ: Angular Momentum and Its Conservation Recall that linear momentum was p = mv A net torque causes a change in angular momentum. Systems that can change their rotational inertia through internal forces will also change their rate of rotation since ang momentum is conserved: If the net torque on an object is zero, the total angular momentum is constant. A student (60kg) sits at the edge of spinning merry-go-round (mass = 100kg, radius = 2.0m) that spins on frictionless axle with speed 2.0rads/s. Student then walks very slowly from edge towards center. Find new angular speed when she reaches 0.50m from center. Must find Isys Conditions for equilibrium 1) 2) 3) Example1 A bridge weighs 2.23x106 N. On it are a tractor-trailer truck that has mass of 14900kg and is positioned 9.75m from the left end of the bridge. A car with mass 1590kg is located 10.80m from the right side of the bridge. A pickup truck (2409kg) is positioned 5.25m from the right side of the bridge. a) What is the upward force exerted by the pier on the right to support the bridge. b) What is the upward force exerted by the pier on the left to support the bridge. Example2 A 75-kg block is suspended from the end of a uniform 100-N beam. If θ = 30º, what are the values of T2 as well as the horizontal and vertical forces on the hinge? Example3 A 700N bear walks on a beam to get a basket of goodies. The uniform beam weighs 200N and is 6.0m long. The goodies weigh 80N. Find the tension in the wire and the components of the reaction force at the hinge when the bear is at x = 1.0m Example4 A ladder of length 8.0m and weight 350N is leaning against a smooth wall at an angle of 60o with the horizontal. A person of mass 90kg stands 3/4 of the way up the ladder. a) What frictional force does the ground need to apply to prevent the ladder from sliding? b) What is the minimum coefficient of static friction? 483.5N, 0.39 Example5 A 2.73-kg lamp is sitting on a table as shown. The lamp has a base with a diameter of 18.0 cm, a height of 68.7 cm. The coefficient of static friction between the base of the lamp and the table is 0.32. A horizontal force, F, is applied to the central column of the lamp at a height of h. What is the maximum height above the base that the force, F, can act without toppling the lamp? Note that pivot has been chosen. Force diagram next slide FN