Review Chp 7: Accounting with Mechanical Energy: the overall Bank Balance When we judge how much energy a system has, we must have two categories: Kinetic energy (K sys ), and potential energy (U sys ). The sum of these two forms of energy (like two sub accounts in a bank balance) is the Mechanical Energy, E mech = K sys + U sys A conservative force acting alone on an object simply exchanges one form of energy to another. So when measuring the change in an object s total mechanical energy, we need only compute what work has been done on it by non-conservative forces: W nc = ΔE mech The non conservative force is like a bank robber, its takes from the total account of energy. Energy problems look like this: An object begins (high/ low/ moving/ not moving/ compressing a spring), and goes (down/ up/ over/ through/ out to lunch/ anywhere!... doesn t matter), encounters/doesn t encounter a frictional surface, and ends (high/ low/ moving/ not moving/ compressing a spring). Find (m, k, x, h, potential, kinetic, or total mechanical energy). Momentum conservation problems 1. For momentum conservation, there must be NO net force. Momentum is a vector, it must be treated in separate directions! 2. Draw a picture showing the system JUST before and JUST after the event Include coordinate system (label axes) Label initial and final velocity vectors 3. Equate initial and final momentum Pay attention to signs! 4. Use given information to solve Use minus signs with velocities if necessary and check units of result Example 8-1 During repair of the Hubble Space Telescope, an astronaut replaces a damaged solar panel during a spacewalk. Pushing the detached panel away into space, she is propelled in the opposite direction. The astronaut s mass is 60 kg and the panel s mass is 80 kg. Both the astronaut and the panel initially are at rest relative to the telescope. After the push the panel is moving at 0.30m/s relative to the telescope. What is her velocity relative to the telescope? Momentum is a VECTOR Example 8-2 A runaway 14,000-kg railroad car is rolling horizontally at 4.00m/s toward a switchyard. As it passed by a grain elevator, 2000kg of grain suddenly drops into the car. How long does it take the car to cover the 500m distance from the elevator to the switchyard? Assume the grain falls straight down and friction is negligible. The condition necessary for the Conservation of Linear Momentum in a given system is that 1. energy is conserved. 2. one body is at rest. 3. the net external force is zero. 4. internal forces equal external forces. 5. None of the above. 1
A golf ball and a Ping-Pong ball are dropped in a vacuum chamber. When they have fallen halfway to the floor, they have the same 1. speed. 2. potential energy. 3. kinetic energy. 4. momentum. 5. speed, potential energy, kinetic energy, and momentum. Two cars of equal mass travel in opposite directions at equal speeds. They collide in a perfectly inelastic collision. Just after the collision, their velocities are 1. zero. 2. equal to their original velocities. 3. equal in magnitude but opposite in direction from their original velocities. 4. less in magnitude and in the same direction as their original velocities. 5. less in magnitude and opposite in direction from their original velocities. Momentum continued: Collisions Want to be a pool shark? Inelastic Collision: Kinetic energy is not conserved. Examples: Explosions (internal force pushes apart), Things sticking together, like putty or car crash (kinetic energy goes into deformation and sticking instead of motion) Elastic Collision: Conserves Kinetic Energy Happens between rigid objects like pool balls (all kinetic energy is returned) Momentum can be conserved while Energy is not conserved Impulse and Average Force When objects collide, they exert large forces on each other for a brief time Impulse is a measure of both the strength and duration of the force Impulse is a vector The force may vary during the time of contact (integral definition on impulse ) The x component of impulse is the area under the curve of an x component force as a function of time Impulse-momentum theorem derivation Units What are the units of Momentum? What are the units of Impulse? 2
Momentum is conserved in which of the following? 1. elastic collisions 2. inelastic collisions 3. explosions 4. collisions between automobiles when friction from the road is negligible 5. All of the above. Example 8-5 With an expert karate blow, you shatter a concrete block. Consider your hand to have a mass of 0.70 kg, to be moving 5.0 m/s as it strikes the block and to stop 6.0 mm beyond the point of contact. a) What impulse does the block exert on your hand? b) What is the approximate collision time and average force the block exerts on your hand? Perfectly inelastic Collisions in one dimension Example 8-9: An astronaut of mass 60 kg is on a space walk. You throw him a repair manual with a speed of 4 m/ s relative to the spacecraft. He is initially at rest relative to the spacecraft before catching the 3.0-kg book. Find: a) his velocity just after he catches the book b) the initial and final kinetic energies of the book- astronaut system c) the impulse exerted by the book on the astronaut Example 8-10 Ballistic Pendulum In a feat of public marksmanship, you fire a bullet into a hanging wood block, which is a device known as a ballistic pendulum. The block, with the bullet embedded, swings upward. Noting the height reached at the top of the swing, you immediately inform the crowd of the bullet s speed. How fast was the bullet traveling? 3-D elastic collision Example 8-16: An object with mass m1 and with an initial speed of 20 m/s undergoes an off-center collision with a second object of mass m2. The second object is initially at rest. After the collision the first object is moving at 15 m/s at an angle of 25 degrees with the direction of the initial velocity of the first object. In what direction is the second object moving? Why is physics important? We observe so we can predict The future of our galaxy 3
Angular Velocity and Angular Acceleration concepts: 1. Use when a rigid object rotates (picture) 2. Arclength revisited 3. One revolution (1 rev) 4. Definition of Angular Velocity 5. Direction is important 1. One way is positive (usually counter clockwise) and one is negative 6. Definition of Angular Acceleration Units of various accelerations Linear Acceleration, a units: m/s 2 Constant Acceleration Kinematics Equations with a Twist (literally) Linear Kinematics Now Rotational Kinematics Centripetal Acceleration, a c units: m/s 2 Tangential Acceleration, a T units: m/s 2 Angular Acceleration, α units: 1/s 2 or rad/s 2 Signs for x, v, a matter! Both assume initial t =0 Signs for θ, ω, α matter! Setting up a rotation problem You must decide on the positive direction (convention is usually counter clockwise) RHR Just like linear, draw a picture, write your knowns Is it possible to have positive angular velocity and negative angular acceleration? Example 9-1 A compact disk rotates from rest to 500rev/min in 5.5s. a) What is its angular acceleration assuming that it is constant? b) How many revolutions does the disk make in 5.5 s? c) How far does a point on the rim 6.0 cm from the center travel during the 5.5 s? 4
Rotational Kinetic Energy Linear and rotational K (a new kinetic energy!) Define Moment of Inertia (this week s lab) An object s resistance to angular acceleration Property of matter (rotating matter) Remember mass is an object s resistance to linear acceleration Moment of Inertia depends on an object s shape, mass and location of axis of rotation Example: A baseball bat s I is less when you choke up on the bat (easier to begin accelerating) Finding Moment of Inertia For a System of Discrete Particles For a Continuous Objects Discrete Particle Example 9-2: An object consists of four point particles, each of mass m, connected by rigid massless rods to form a rectangle of edge lengths 2a and 2b. The system rotates with angular speed ω about an axis in the plane of particles. Find the kinetic energy of this object Continuous Object Example 9-4: Find the moment of inertia of a thin uniform rod of length L and mass M about an axis perpendicular to the rod and through one end. Hoop Disk Other Integral Examples Need Parallel Axis Theorem for objects rotating off-center Example 9-5: Find the moment of Inertia of a rod, mass M, length L, rotating around it s center (not it s end). 5