When does a force do positive work?
When does a force do zero work?
When does a force do negative work?
Units of Work
SI System:
British System:
cgs System:
Atomic Level:
Problem: A droplet of water of mass 50 mg falls at constant speed under the influence of gravity and air resistance. After the drop has fallen 1.0 km, what is the work done by a) gravity and b) air resistance?
Problem: A sled loaded with bricks has a mass of 20.0 kg. It is pulled at constant speed by a rope inclined at 25o above the horizontal, and it moves a distance of 100 m on a horizontal surface. If the coefficient of kinetic friction between the sled and the ground is 0.40, calculate:
a) The tension in the rope.
b) The work done by the rope on the sled
c) The work done by friction on the sled.
Dot product equation for work:
How to evaluate the dot product for work if given magnitudes and directions of vectors:
How to evaluate the dot product for work if given unit vector representation of vectors:
Applications of the dot product:
When is the dot product biggest?
Problem: Vector A has a magnitude of 8.0 and vector B has a magnitude of 12.0. The two vectors make an angle of 40o with each other. Find A•B.
Problem: A force F = (5.0i + 6.0j – 2.0k)N acts on an object that undergoes a displacement of r = (4.0i – 9.0j + 3.0k)m. How much work was done on the object by the force?
Problem: A force F = (5.0i – 3.0j) N acts upon a body which undergoes a displacement d = (2.0i – j) m. How much work is performed, and what is the angle between the vectors?
Formula for calculating work for infinitesimal displacement:
Integral expression for work when force is variable with position:
Problem: A force acting on a particle is Fx = (4x – x2)N. Find the work done by the force on the particle when the particle moves along the x-axis from x= 0 to x = 2.0 m.
Problem: Derive an expression for the work done by a spring as it is stretched from its equilibrium position
Problem: How much work does an applied force do when it stretches a nonlinear spring where the force varies according to the expressions F = (300 N/m) x – (25 N/m2) x2 from its equilibrium length to 20 cm?
Define what is meant by “net work”
State the work-energy theorem in equation form:
Give an equation for kinetic energy:
Problem: A force of F1 = (4.0 i + j) N and another of F2 = -4.0 j N act upon a 1 kg object at rest at the origin. What is the speed of the object after it has moved a distance of 3.0 m?
Define power:
Give several equations for calculating power:
Give the SI unit of power:
Problem: A 1000-kg space probe lifts straight upward off the planet Zombie, which is without an atmosphere, at a constant speed of 3.0 m/s. What is the power expended by the probe’s engines? The acceleration due to gravity of Zombie is ½ that of earth’s.
Problem: Develop an expression for the power output of an airplane cruising at constant speed v in level flight. Assume that the aerodynamic drag force is given by FD = bv2.
By what factor must the power be increased to increase airspeed by 25%?
Problem: Using what you know about units, calculate how many Joules is in a kilowatt-hour.
Conservative forces:
Work in moving an object is path independent.
Work in moving an object along a closed path is zero.
Work is directly related to a negative change in potential energy
Ex: gravity, electrostatic, magnetostatic, springs
Non-conservative forces:
Work is path dependent.
Work along a closed path is NOT zero.
Work may be related to a change in mechanical energy, or thermal energy
Ex: friction, drag, magnetodynamic
Define potential energy:
Give examples of potential energy:
How does potential energy relate to work?
Gravitational potential energy:
Spring potential energy:
Problem: Three identical springs (X, Y, and Z) are hung as shown. When a 5.0-kg mass is hung on X, the mass descends 4.0 cm from its initial point. When a 7.0-kg mass is hung on Z, how far does the mass descend?
Problem: Gravitational potential energy for a body a large distance r from the center of the earth is defined as shown below. Derive this equation from the Universal Law of Gravity.
Law of Conservation of Mechanical Energy
Problem: A single conservative force of F = (3i + 5j) N acts on a 4.0 kg particle. Calculate the work done if the particle if the moves from the origin to r = (2i - 3j) m. Does the result depend on path?
What is the speed of the particle at r if the speed at the origin was 4.0 m/s?
What is the change in potential energy of the system?
Sample Problem: A bead slides on the loop-the-loop shown. If it is released from height h = 3.5 R, what is the speed at point A? How great is the normal force at A if the mass is 5.0 g?
General statement of the Law of Conservation of Energy (including dissipative forces).
Sample Problem: A 2,000 kg car starts from rest and coasts down from the top of a 5.00 m long driveway that is sloped at an angel of 20o with the horizontal. If an average friction force of 4,000 N impedes the motion of the car, find the speed of the car at the bottom of the driveway.
Problem: A parachutist of mass 50 kg jumps out of a hot air balloon 1,000 meters above the ground and lands on the ground with a speed of 5.00 m/s. How much energy was lost to friction during the descent?
What relationship exists between force and potential energy?
Stable Equilibrium:
Unstable Equilibrium
Neutral Equilibrium
Equation that relates force and potential energy:
Stable equilibrium equations:
Unstable equilibrium equations:
Neutral equilibrium equations:
Problem: The potential energy of a two-particle system separated by a distance r is given by U(r) = A/r, where A is a constant. Find the radial force F that each particle exerts on the other.
Problem: A potential energy function for a two-dimensional force is of the form U = 3x3y – 7x. Find the force acting at a point (x,y).
12/7/2008 6 Bertrand
12/7/2008 6 Bertrand