Chapter 7: Energy of a System

NEWTON’S LAWS OF MOTION! THE COURSE THEME is NEWTON’S LAWS OF MOTION! Chs. 5 & 6: Motion analysis with forces. NOW (Chs. 7 & 8): An alternative analysis using the concepts of Work & Energy. Easier? My opinion is yes! Conservation of Energy: NOT a new law! We’ll see that this is just Newton’s Laws of Motion re-formulated or re-expressed (translated) from Force Language to Energy Language.

We’ve expressed Newton’s Laws of Motion using concepts of position, displacement, velocity, acceleration, force. Newton’s Laws with Forces: Quite general (macroscopic objects). In principle, could be used to solve any dynamics problem, But, often, they are very difficult to apply, especially to very complicated systems. So, alternate formulations have been developed. Often easier to apply. One of these is an approach that uses Energy instead of Force as the most basic physical quantity. Newton’s Laws in a different language (Energy). Before we discuss these, we need to learn vocabulary in Energy Language . Energy: Very common term in everyday usage. Everyday meanings might not coincide with the PHYSICS meaning! Every physical process involves energy or energy transfer or transformations. Energy in physics can be somewhat abstract.

Sect. 7.1: Systems & Environments So far, we’ve expressed Newton’s Laws of Motion in terms of forces & we’ve considered the dynamics properties of a particle by talking about various particle properties. Now, we take a different approach & talk about Systems & System Properties. System: A small portion of the universe which we focus on in a given problem. What the system is depends on the problem. A System may be, for example: Single particle. Collection of particles. A region of space. May vary in size & shape, depending on the problem In addition to a System, we also talk about the system environment. System interacts with environment at it’s boundaries.

Sect. 7.2: Work Done by a Constant Force Work: Precisely defined in physics. Describes what is accomplished by a force in moving an object through a distance. For an object moving under a Constant Force, Work done (W) = product of magnitude of displacement (d)  component of force parallel to displacement (F||): W = F||Δr = FΔr cosθ

Work by a Constant Force W = F||Δr = FΔr cosθ Δr NOTE: Valid for a constant force ONLY!

W = F||Δr = FΔr cosθ Simple special case: F & Δr are parallel: θ = 0, cosθ = 1  W = Δr d Example: Δr = 50 m, F = 30 N W = (30N)(50m) = 1500 N m Work units: Newton - meter = Joule 1 N m = 1 Joule = 1 J

Can exert a force & do no work! Could have d = 0  W = 0 W = F||d = Fd cosθ Can exert a force & do no work! Could have d = 0  W = 0 Could have F  d  θ = 90º, cosθ = 0  W = 0 Example, walking at constant v with grocery bag:

W = F||Δr = FΔr cosθ Typical Problem: Normal force, θ = 90°, cosθ = 0 An object displaced by force F on a frictionless, horizontal surface. Free body diagram is shown  Normal force n & weight mg do no work in the process, since both are perpendicular to the displacement. Normal force, θ = 90°, cosθ = 0 Weight: θ = 270 (or - 90°), cosθ = 0

W = F||Δr = FΔr cosθ Note: W is a scalar (contrast to forces, which are vectors). However, W can have either a positive or a negative sign, since cosθ can be positive or negative. IMPORTANT: Work (as we’ll see) is a Transfer of Energy: The System either gains energy (if W > 0) or loses energy (W < 0).

Example 7.1

Example m = 50 kg, FP = 100 N, Ffr = 50 N, θ = 37º W = F||Δr =FΔr cosθ