Energy and Work
Work done by a Force Distance x Force x Distance x Force x Distance x Force x Textbook defines work due to a force as the product of the force on an object and the displacement of that object in the same direction as the force. For an object that only moves in the x direction, we have We can make a graph of Force vs. Displacement in one dimension:
Work done by a Force Textbook defines work due to a force as the product of the force on an object and the displacement of that object in the same direction as the force. For an object that only moves in the x direction, we have In general, force and displacement are vectors. Thus, the most general expression for work is given by Where θ is the angle picture below. Picks off the component of F parallel to Δx.
Positive and Negative Work The direction of the force compared with that of the displacement determines if energy is added or subtracted from a system. W = Fd cos(0) Maximum energy added W = Fd cos( less than max energy added W = Fd cos(90) Zero energy added W = Fd cos(180) Maximum energy subtracted W = Fd cos( less than max energy subtracted
Work and Kinetic Energy Let’s consider one-dimensional motion: Definition 1D Kinematics Rearrange Substitute Kinetic Energy!
Force at an Angle Δh=3 m m=10 kg If the object rolls down the ramp, then the work that is done by gravity is: θ=30° If the object falls off of the ramp, then the work that is done by gravity is: The angle θ has no effect on the total work done (neglecting friction)!
Gravitational Potential Energy Δh=3 m m=10 kg θ=30° The work done by gravity is equal to negative the change in the gravitational potential energy, U g. Only care about change in U g, so location of U g = 0 is arbitrary!
Spring Potential Energy m -F s xx kxkx From Hooke’s Law:
Energy Conservation If W nc = 0, and there are no other conservative forces present, we have conservation of energy. Work/KE Theorem Label specific conservative forces Definition of potential energies
The Mean Streak (Cedar Point) What is theoretical top speed? Maximum height: 160 ft (49 m) Top speed (measured): 65 mph (29 m/s) What about using kinematics instead? We don’t know the path, but let’s make one up…
Example A 400 kg cart traveling at 3 m/s rolls down an incline and encounters a spring with a spring constant of 1000 N/m. How far does the spring compress? What is the speed of the cart just before striking the spring? 3 m/s 0.2 m
Example m A 400 kg block starts at rest against a spring with k = 1000 N/m that has been compressed by 1.19 m. What is its speed v t at top? So what height can it reach?
Reminders Exam tomorrow at 9:35 (Osmond 119) – Bring a calculator (NOT your phone), pencil, eraser – Equation sheet provided Reading: Chapter 5.5 – 5.7, 6.1 – 6.3 HW 04 Due MONDAY at 10 PM – Additional office hours Monday at 11:00 AM