8 01 Final Exam Review Sheet 8 01 Review Exam 1 Newton s First Law law of inertia Newton s Second Law F ma Newton s Third Law equal and opposite forces F F Average velocity V Instantaneous velocity on a position vs time graph the instantaneous velocity is equal to the slope of the line tangent to the curve Average acceleration a Instantaneous acceleration on a velocity vs time graph the instantaneous acceleration is equal to the slope of the line tangent to the curve Change in velocity integral of acceleration v t v a t t Change in position integral of velocity x t x v t t Direction of a vector arctan Magnitude of a vector A Tension is uniform in a massless rope Tension in a massive rope now nd the tension in the rope as a function of distance from the block Now take an imaginary slice a distance x away from the end of the rope such that and and then apply newton s second law Tension of a suspended massive rope tension at the top is equal to the gravitational force and the tension at the bottom is zero Centripetal force a force that stays constant in magnitude but always points towards the center of the circle Hooke s law stretching or compressing the spring by different amounts produces different acceleration Restoring force a force where the direction of acceleration is always towards the equilibrium position Kinetic friction opposes motion Static friction varies in direction and magnitude Just slipping if the magnitude of the applied force along the direction of the contact surface exceeds the magnitude of the maximum value of static friction then the object will start to slip 8 01 Review Exam 2 Acceleration will always have a non positive radial component centripetal acceleration due to the change in direction of velocity Acceleration may also have a tangential component Centripetal acceleration a Momentum p mv Impulse I p F t F t The total external force is equal to the rate of change of the momentum of the system F p t If there is no external force there is no change in momentum If the collision time is instantaneous then the change in momentum is approximately zero Position of center of mass R Velocity of center of mass V No matter where force is applied to an object the acceleration of the center of mass will be the same regardless of the application because acceleration is independent Relative velocity of a person jumping off a cart with speed u v u v In a rocket ejecting fuel problem chose a reference frame in which the rocket is at rest at the time t so the second momentum diagram is for the rect and fuel at time t t Rocket equation v If a rocket burns fuel at a constant rate the acceleration of the rocket is increasing Kinetic energy k 1 2mv Work done by a constant force W F x The work done by static friction to accelerate a persons feet forward is zero because when the foot makes contact with the ground there is no displacement Total work done is equal to the area bounded by the graph of the function F x between the initial and nal positions Work done along an arbitrary path W F r Conservative force is path independent where as non conservative force is path dependent Potential energy difference U W F r Potential energy of a spring U x 1 2kx x component of the force U x kx The magnitude of the force is greatest when potential energy equals mechanical energy The kinetic energy is greatest when potential energy is minimized The velocity is zero when potential energy equals mechanical energy because kinetic energy is equal to zero The force is zero when the negative derivative of potential energy is zero 8 01 Review Exam 3 Torque equation Rotational kinetic energy about axis passing though axel K Parallel axis theorem I I md If the momentum of inertia increases the angular acceleration decreases Problem solving strategy xed axis rotation Draw free body diagrams and indicate the point of application of each force Select a point to compute torque about usually the center of mass Choose a coordinate system Apply newton s second law and toque law to obtain equations Look for constraint condition between rotational acceleration and linear acceleration Angular momentum L r x mv Torque is equal to the time derivative of the angular momentum When the torque about s is zero angular momentum is constant L L A rigid body with rotational symmetry rotates at a constant angular speed about it symmetry z axis L is constant A non symmetric body that rotates with constant angular acceleration relative to the origin L is constant but L L is not Kinetic energy of rotation and translation K Calculating speed of blocks in a pulley system using conservation of energy for xed axis rotation Choose zero for the gravitational potential energy at a height equal to the center of the pulley Solve for the initial mechanical energy which is all potential Solve for the nal mechanical energy when the block has moved a distance d x Use the tangential component of velocity v R Because energy is conserved set the energies equal to each other Calculating the magnitude of acceleration of blocks in a pulley system using torque equations Find the torque about the center of the pulley Draw the free body diagrams and set up force equations Use in your equation for friction N mg Set up an equation for the tangential acceleration a R Calculating the moment of inertia and angular speed of a physical pendulum Solve for the moment of inertia using the parallel axis theorem Plug that value in to get the moment of inertia of the system Finally sole to get the moment of inertia about the center of mass Use the conservation of mechanical energy to nd the angular speed of the pendulum at the bottom of its swing Calculating the impact parameter of a meteor yby Set up the force of the meteor Calculate the initial angular momentum Calculate the nal angular momentum Set the two equal to each other to get the nal velocity and solve for h Simple harmonic motion SMH example equation y t Asin Simple harmonic oscillator SHO equation Acos t Acos t Angular velocity Period T 2 Equation of motion Simple harmonic oscillator SHO general solution x t Ccos t Dsin t where C x and D Calculating the period and velocity of a block spring system Substitute the initial conditions into x t to get the position Substitute the initial conditions into v t to get the velocity Find the time t when the block rst reaches equilibrium and use the equation t to solve for the time in term of T Plug t into the velocity equation and simplify Simple
View Full Document
Unlocking...