Section 4.9: Review Examples
These examples consolidate the concepts of work, energy, and energy transformations discussed in Units 4.1–4.8. Focus on applying formulas and interpreting energy bar charts.
Example 1
A 2 kg block slides down a 5 m high frictionless ramp. Calculate the velocity at the bottom and draw an energy bar chart at top, middle, and bottom.
Energy conservation: \( mgh = \frac{1}{2}mv^2 \Rightarrow v = \sqrt{2gh} = \sqrt{2*9.8*5} \approx 9.9\text{ m/s} \)
Energy bar chart: Top: PE = 98 J, KE = 0; Middle: PE = 49 J, KE = 49 J; Bottom: PE = 0, KE = 98 J
Example 2
A 1 kg pendulum bob is released from 0.8 m height. Find its speed at the lowest point and represent energies in a bar chart.
Energy: \( mgh = \frac{1}{2}mv^2 \Rightarrow v = \sqrt{2gh} = \sqrt{2*9.8*0.8} \approx 3.96\text{ m/s} \)
Bar chart: Highest: PE = 7.84 J, KE = 0; Halfway: PE = 3.92 J, KE = 3.92 J; Lowest: PE = 0, KE = 7.84 J
Example 3
A 3 kg block compresses a spring by 0.2 m (k = 200 N/m). Determine maximum speed if released on frictionless surface.
Spring potential energy: \( \frac{1}{2}kx^2 = \frac{1}{2}*200*0.2^2 = 4\text{ J} \)
Convert entirely to KE: \( \frac{1}{2}mv^2 = 4 \Rightarrow v = \sqrt{2*4/3} \approx 1.63\text{ m/s} \)
Practice Problems
- Block of mass 5 kg slides down 8 m high ramp, frictionless. Find velocity at bottom and draw energy bar charts.
- Spring (k = 500 N/m) compressed 0.1 m, mass = 2 kg. Determine maximum speed and energy distribution.
- Ball dropped from 10 m. Sketch energy bar charts at 0 m, 5 m, and impact points.
- Pendulum length 1 m, bob mass 1.5 kg. Released from 0.5 m height. Find speed at bottom and chart energies.
- Roller coaster car, 600 kg, height 15 m. Compute speed at lowest point ignoring friction and draw bar chart.