Conservation Laws Work and Conservation of Energy

Conservative Forces Act Work of Conservative forces Conservative Forces The force of gravity, Fg , and the restoring force, FS , are conservative forces. When these forces act, energy is conserved in the object or system under investigation. Forces from the environment (objects or systems outside of those under investigation) are not involved. Environment Object / System K Conservative Forces Act Ug US Example: A ball dropped from a height h loses gravitational potential energy. If there is no environmental interference, such as air resistance, the resulting kinetic energy will have the same magnitude as the potential energy that was lost. Force gravity is a conservative force.

Non-Conservative Forces Act Conservative Forces Act Work of Non-Conservative forces Non-conservative Forces Applied forces, F , and friction, f , are examples of non-conservative forces. When these forces act, energy enters or leaves the system as the work of non-conservative forces. Energy in the system is no longer conserved. However, energy in the universe remains the same. Energy gained/lost by the object/system is lost/gained by the environment. Environment Object / System −W +W Non-Conservative Forces Act K Conservative Forces Act Ug US Example: When an object with kinetic energy encounters a rough surface the work of friction, Wf , drains energy from the system. The lost energy transfers to the environment. The surface becomes warmer.

Any non-conservative force ? Rough surface: Friction Example 1 A 2.0 kg mass is initially moving at 4.0 m/s along a rough horizontal surface, with μk = 0.20 . How far will the object travel before coming to a stop? m N Fg f v m N Fg Final h0 = 0 x0 = 0 (no spring) v0 = 0 m/s d = ? Any non-conservative force ? Rough surface: Friction Initial h0 = 0 x0 = 0 (no spring) v0 = 4.0 m/s

Conservation of energy revisited Use the overall concept equation as a guide to building a custome equation Select from the above menu of available energies to build your equations.

Example 2 Initial Conditions? Non-conservative forces? A ball is thrown upward from the ground. How high will it go? Initial Conditions? Starts at lowest point, h0 = 0 No spring, x0 = 0 To move upward there must be an initial velocity, v0 Non-conservative forces? None mentioned Final Conditions? Reaches maximum height, h Still no spring, x = 0 At max height objects have an instantaneous zero velocity, v0

Example 3 Initial Conditions? Non-conservative forces? A toy gun consists of a spring in a tube. The compressed spring launches a ball horizontally. Determine the ball’s speed when it exists the tube. Initial Conditions? Starts at lowest point, h0 = 0 The spring is compressed, x0 Initially at rest, v0 = 0 Non-conservative forces? None mentioned Final Conditions? Moves horizontally, h = 0 Spring restored, x = 0 Ball is now moving, v

Example 4 Initial Conditions? Non-conservative forces? A mass initially at rest slides down a rough incline. Determine the speed of the mass after sliding a distance, Δr , measured along the incline. Initial Conditions? Starts at top of incline, h0 No spring, x0 = 0 Initially at rest, v0 = 0 Non-conservative forces? Friction on an incline Final Conditions? Reaches lowest point, h = 0 No spring, x = 0 Determine speed, v Be careful. Students solving for the work of friction often focus on getting the force of friction correct, and as a result they forget to include the displacement, d .

Example 5 A 2.0 kg mass starts from rest at the top of a smooth, frictionless, quarter circle of R = 3.0 m. When it reaches the bottom of the curve it moves along a horizontal section of track that is rough, μk = 0.50 . Determine the horizontal displacement, d , of the mass before coming to rest. This above equation is a generic equation using standard variables. Adapt the variables to fit the problem being solved. d R

Example 6 A 0.50 kg mass starts from rest at the top of a smooth, frictionless, quarter circle of R = 1.5 m. When it reaches the bottom of the curve the mass travels 2.0 m on a rough horizontal section, μ = 0.35 . At the end of the track there is a 0.80 m drop. Determine the horizontal displacement x , during the drop. This is actually two separate problems 2.0 m 0.80 m R x = ? 1st solve the energy problem to find the speed that it has at the end of the horizontal rough surface. v Then solve for a horizontally launched projectile.

Example 6 A 0.50 kg mass starts from rest at the top of a smooth, frictionless, quarter circle of R = 1.5 m. When it reaches the bottom of the curve the mass travels 2.0 m on a rough horizontal section, μ = 0.35 . At the end of the track there is a 0.80 m drop. Determine the horizontal displacement x , during the drop. 2.0 m R Adapt the variables to fit the problem v

Example 6 Horizontally launched projectile A 0.50 kg mass starts from rest at the top of a smooth, frictionless, quarter circle of R = 1.5 m. When it reaches the bottom of the curve the mass travels 2.0 m on a rough horizontal section, μ = 0.35 . At the end of the track there is a 0.80 m drop. Determine the horizontal displacement x , during the drop. Horizontally launched projectile Use speed found using conservation of energy 0.80 m x = ? v

Example 7 A 5.0 kg mass initially at rest slides down a smooth, frictionless, quarter circle of R = 4.0 m. Then the mass moves 3.0 m on a rough horizontal section of track, μk = 0.10 . Finally the mass compresses a spring with a spring constant of k = 20 N/m . Determine maximum compression of the spring. 3.0 m R Adapt the variables to fit the problem