Section 4.2: Energy Conservation, Loss, and Conversion

In real-world scenarios, mechanical energy is often converted to other forms or partially lost due to non-conservative forces like friction. Understanding these transformations helps in solving energy-related problems.

        Conservation of Mechanical Energy (ideal): 
        \[ ME_{initial} = ME_{final} \Rightarrow KE_i + PE_i = KE_f + PE_f \]
      

        Non-conservative forces: Work done by friction or air resistance reduces mechanical energy: 
        \[ W_{nc} = \Delta ME = ME_{final} - ME_{initial} \]
      

        Energy Conversion Examples:
        Potential → Kinetic (falling objects)
Kinetic → Thermal (friction)
Mechanical → Electrical (generators)

      

Example 1

A 2 kg block slides down a 5 m high frictionless ramp. Find speed at the bottom.

Using energy conservation: \( m g h = \frac{1}{2} m v^2 \Rightarrow v = \sqrt{2 g h} = \sqrt{2 \cdot 9.8 \cdot 5} \approx 9.9 \text{ m/s} \)

Example 2

A 3 kg box is pushed across a horizontal surface 4 m with a frictional force of 10 N. Find the work done by friction and the final mechanical energy if it started from rest.

Work done by friction: \( W_f = F d \cos 180^\circ = -10 \cdot 4 = -40 \text{ J} \)

Initial mechanical energy \( ME_i = 0 \) → Final \( ME_f = ME_i + W_{nc} = 0 - 40 = -40 \text{ J} \) (energy lost to heat)

Practice Problems

A 1.5 kg object slides down a 3 m high frictionless hill. Find speed at bottom.
A car engine delivers 5000 J while friction dissipates 1200 J. What is net mechanical energy gain?
Calculate speed of a pendulum at lowest point, starting from 2 m height, ignoring air resistance.
A sled slides down a snowy slope 10 m long with friction 50 N. Mass of sled 20 kg. Find work done by friction.
A roller coaster car of mass 250 kg starts from rest at 15 m height. Friction causes 3000 J energy loss. Find speed at bottom.

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