Work Done by a Constant Force The work done by a constant force is defined as the distance moved multiplied by the component of the force in the direction of displacement: Chapter 6: Work and Energy In the SI system, the units of work are joules: As long as the force exerted on the object is not displacing it, this force is doing no work on the object. The work can be either positive or negative, depending on the angle between the force and the displacement.

Draw a free-body diagram. 1. Choose a coordinate system. 2. Apply Newton’s laws to determine any unknown forces. 3. Find the work done by a specific force. 4. To find the net work, either find the net force and then find the work it does, or find the work done by each force and add. Solving work problems

Work done by forces that oppose the direction of motion, such as kinetic friction, will be negative, since Centripetal forces do no work, as they are always perpendicular to the direction of motion, so

Work Done by a Varying Force For a force that varies, the work can be approximated by dividing the distance up into small pieces, finding the work done during each, and adding them up. As the pieces become very narrow, the work done is the area under the force vs. distance curve.

Kinetic Energy, and the Work-Energy Principle Mechanical energy is defined as the ability to do work. There are two varieties of mechanical energies. Kinetic energy is energy of motion. It depends only on mass and velocity. Potential energy is energy associated with forces that depend on the positions or configurations (relative positions) of objects. If we write the acceleration in terms of the velocity and the distance, we find that the work done here is We define the kinetic energy: Varieties of Mechanical Energy

Kinetic Energy

Hence the work done is equal to the change in the kinetic energy during the time interval D t = t 2 - t 1 : If the net work done on the object is positive, its kinetic energy increases. If the net work done on the object is negative, its kinetic energy decreases. Because work and kinetic energy can be equated, they must have the same units. In the SI system, kinetic energy is measured in joules. We found that the work done by the net force on the object is Let us define the kinetic energy of the object at any time as Kinetic energy depends only on object’s mass and velocity (which is a property of motion), but not on its position. This is called the Work-Kinetic Energy Theorem.

In raising a mass m to a height h, the work done by the external force on the system (mass + Earth) is Mechanical energy is defined as a property of an object at specified time and position relative to its surroundings. There are two varieties of mechanical energies. Kinetic energy is energy of motion (depends only on mass and velocity). Potential energy is energy associated with forces that depend on the position relative to the surroundings. Only changes in its mechanical energy can be measured by work done on an object by net force, so energy of an object is known only up to a constant. Gravitational Potential Energy on Earth We therefore define the gravitational potential energy:

If, where do we measure y from? Since only changes in potential energy can be measured, it does not matter, as long as we are consistent about where we choose y = 0. The net force (sum of external and gravitational forces) is zero and thus does no work as the object’s is raised, so its kinetic energy is unchanged. Gravitational potential energy is defined for a system (Earth + object): depends on the gravitational force exerted by the Earth on the object and on object’s height relative to Earth surface. It will be different on the Moon. It does not depend on the object’s velocity. The drop causes transformation of stored potential energy into kinetic energy.

Elastic Spring Potential Energy Potential energy can also be stored in a spring when it is compressed; releasing the spring transforms it into kinetic energy. Hooke’s Law The force required to compress or stretch a spring is: where k is called the spring constant, and needs to be measured for each spring.

The force required to compress or stretch a spring is: where k is called the spring constant, to be measured for each spring). The magnitude of the force increases as the spring is stretched or compressed further. We find that the potential energy of compressed or stretched spring, measured from its equilibrium position, can be written: Other Forms of Energy Electric energy, nuclear energy, thermal energy, chemical energy.

Work by conservative forces changes potential energy of the system: If net force on each object is conservative, system’s mechanical energy is conserved: Nonconservative forces acting on system’s objects msy be of two types: External forces (exerted by external objects) Dissipative forces (internal force between system’s objects) Mechanical energy of a system may change due to work done by nonconservative forces:

Energy Conservation with Dissipative Processes If there are dissipative forces between the system’s objects, such as friction, where do the kinetic and potential energies of each object go? They are transformed into thermal energy, which is a mechanical energy associated with the random components of the velocities and the positions of the microscopic constituents of each object (e.g. atoms). Thermal energy can be calculated, but requires abandoning the particle model of macroscopic objects. It is proportional to system’s temperature. Particle Model of Macroscopic Object Microscopic Model of Macroscopic Object

Dissipative Forces Kinetic friction and drag forces between system’s objects are always directed opposite to displacement. Since work done by such forces is always negative over any displacement interval, it cannot be zero over a closed path. Hence if such internal forces are present, the work depends not only on starting and ending points, but also on path taken. Such forces are called dissipative nonconservative forces.

Work done by Dissipative Forces It is always negative, It raises the thermal energy

Law of Conservation of Energy

Problem Solving Using Conservation of Energy In the image on the left, the total mechanical energy is: Neglecting the force of air resistance, the mechanical energy of the system (rock + Earth) is conserved:

If there is no friction, the speed of a roller coaster will depend only on its height compared to its starting height. For a system of (mass + spring) in the absence of external and nonconsevative forces, conservation of energy tells us:

Energy Conservation with Dissipative Processes: Solving Problems Problem Solving: 1. Draw a picture. 2. Determine the system for which energy will be conserved. 3. Figure out what you are looking for, and decide on the initial and final positions. 4. Choose a logical reference frame. 5. Apply conservation of energy. 6. Solve.

Power Power is the rate at which work is done – The difference between walking and running up these stairs is power – the change in gravitational potential energy is the same. In the SI system, the units of power are watts:

Power Power is also needed for acceleration and for moving against the force of gravity. The average power can be written in terms of the force and the average velocity: