Work and Energy ability to do work.

Work and Energy

• Energy and Momentum provide an alternative approach to solving problems related to forces and motion.

• Energy and momentum are conserved quantities, making their conservation laws valuable, especially in systems with many objects where detailed force considerations are impractical.

• These laws apply to various phenomena, including the atomic and subatomic levels where Newton's laws may not be applicable.

• Work and Energy are scalar quantities (no direction).

Work Done by a Constant Force

• Work is defined as the product of displacement magnitude and the component of force parallel to the displacement.

• Equation: $W = F_{\parallel} d$

  • $W$ = Work done

  • $F_{\parallel}$ = Component of constant force parallel to displacement

  • $d$ = Magnitude of displacement

• Alternative equation: $W = Fd \cos{\theta}$

  • $F$ = Magnitude of constant force

  • $d$ = Magnitude of displacement

  • $\theta$ = Angle between force and displacement direction

Special Cases

• Case i: Motion and force are in the same direction ($\theta = 0$ and $\cos{\theta} = 1$).

  • Example: Pushing a cart horizontally with 30 N force over 10 m.
    $W = (30 \text{ N})(10 \text{ m}) = 300 \text{ J}$

  • Define Joule (J) as the unit of work and energy.

• Case ii: Motion and force are perpendicular ($\theta = 90^\circ$ or $270^\circ$, $\cos{\theta} = 0$).

  • Example: Holding a bag of groceries and walking horizontally.

  • No work is done as there's no force in the direction of motion.

• Case iii: Motion and force are opposite (like friction, $\cos{\theta} = -1$).

  • Negative work is done by friction.

• Important to specify:

  • Whether work is done by or on a specific object.

  • Whether work is done by a particular force or the net force.

Example 1

A person pulls a 50-kg crate 40 m along a horizontal floor with a constant force at a 37° angle.
The floor exerts a friction force $F_{fr}$.

• Determine:

  • a) Work done by each force on the crate.

  • b) Net work done on the crate.

• Solution:

  • Work done by normal force ($F_N$) and gravitational force ($mg$) is zero (right angles to displacement).

  • Work done by applied force ($F<em>p$): $W</em>F = F_p d \cos{\theta}$

  • Work done by friction force ($F_{fr}$) is negative.

  • b) Net work done:
    $W<em>{net} = \sum W</em>i = W<em>F + W</em>{fr}$

  • Net work is also calculated by determining the net force and its component along the displacement.

• Work is negative when the force (or its component) opposes motion, causing the object to slow down.

• Positive work indicates the force is speeding up the object.

• Read Pg.199(1-4) and Pg.202(5-*8) from the textbook.

Example 2 (Optional)

Work on a backpack.

• a) Work a hiker does on a 15.0-kg backpack to carry it up a hill of height h.

• b) Work done by gravity on the backpack.

• c) Net work done on the backpack (assuming smooth, constant velocity motion).
  Does the earth do work on the Moon?

The Moon revolves around the earth in a nearly circular orbit, maintained by Earth's gravitational force.

• Does gravity do:

  • a) positive work

  • b) negative work

  • c) no work on the moon?

Work Done by a Varying Force (Graphical Interpretation)

• As a rocket moves away from Earth, work is done against the varying gravitational force.

• Similarly, spring force varies with stretch or compression.

• Work done by a varying force can be determined graphically:

  1. Plot $F_{\parallel}$ (= F cos θ), the component of F parallel to d, versus distance d.

  2. Divide the distance into small segments Δd.

  3. For each segment, indicate the average $F_{\parallel}$ by a horizontal dashed line.

  4. Work done in each segment is ΔW = $F<em>{\parallel}$Δd, the area of a rectangle (Δd wide and $F</em>{\parallel}$ high).

  • Total work is the sum of these areas.

  1. More segments (smaller Δd) increase accuracy, approaching the area under the curve as Δd approaches zero.

• Bottom Line: Work done by a variable/constant force between two points equals the area under the $F_{\parallel}$ vs. d curve between those points.

Example

• A 24 kg rocket car is initially at rest on a frictionless horizontal surface.

• The engine is ignited and the graph shows thrust force, F, versus distance travelled, d, for the rocket car.

• Find the rocket car's speed after it has travelled 200 m.

  $\text{area bounded by graph and d - axis} = \frac{1}{2} \cdot \text{base} \cdot \text{height}$

  $= \frac{1}{2} \times 200 \times 320$

  $= 32 000 \text{ J}$

  Work-energy Theorem: $W = \Delta KE$

  $\Delta KE = KE<em>2 - KE</em>1 = 32 000$

  $KE_1 = 0$

  $KE_2 = \frac{1}{2} \cdot m \cdot v^2$

  $m = 24 \text{ kg}$

  $\frac{1}{2} \cdot m \cdot v^2= 32000$

  $v^2 = \frac{2 \cdot 32000}{m}$

  $v = \sqrt{\frac{2 \cdot 32000}{24}} = 52 \frac{m}{s}$

Work-Energy Principle

• Energy is the