Free Body Diagram Examples for Energy Conservation Problems

📚 Quick Study Guide

• 🔍 Free Body Diagram (FBD): A visual representation of all forces acting on an object. Crucial for analyzing energy conservation.
• ➡️ Energy Conservation Principle: In a closed system, the total energy remains constant. Expressed as: $E_{initial} = E_{final}$
• 📐 Potential Energy (PE): Energy due to position or configuration. Gravitational PE: $PE = mgh$, Spring PE: $PE = \frac{1}{2}kx^2$
• 🏃 Kinetic Energy (KE): Energy due to motion. $KE = \frac{1}{2}mv^2$
• 🔥 Work Done by Non-Conservative Forces: If non-conservative forces (like friction) are present, $E_{initial} + W_{non-conservative} = E_{final}$
• 💡 Steps for Solving Energy Conservation Problems with FBDs:
  1. Draw the FBD.
  2. Identify initial and final states.
  3. Write the energy conservation equation.
  4. Solve for the unknown.

🧪 Practice Quiz

1. A block of mass $m$ slides down a frictionless inclined plane of height $h$. What is its speed at the bottom?
   1. $\sqrt{gh}$
   2. $\sqrt{2gh}$
   3. $2gh$
   4. $mgh$
2. A spring with spring constant $k$ is compressed by a distance $x$. What is the potential energy stored in the spring?
   1. $kx$
   2. $\frac{1}{2}kx$
   3. $kx^2$
   4. $\frac{1}{2}kx^2$
3. A ball of mass $m$ is thrown upwards with an initial velocity $v$. What is the maximum height it reaches (ignoring air resistance)?
   1. $\frac{v^2}{g}$
   2. $\frac{v}{2g}$
   3. $\frac{v^2}{2g}$
   4. $\frac{2v^2}{g}$
4. A block slides down an inclined plane with friction. The work done by friction is $W_f$. If the initial potential energy is $PE_i$ and the final kinetic energy is $KE_f$, what is the relationship between them?
   1. $PE_i = KE_f$
   2. $PE_i = KE_f + W_f$
   3. $PE_i + W_f = KE_f$
   4. $PE_i = KE_f - W_f$
5. A pendulum of length $L$ is released from an angle $\theta$ with the vertical. What is its speed at the bottom of the swing?
   1. $\sqrt{gL(1 - \cos\theta)}$
   2. $\sqrt{2gL(1 - \cos\theta)}$
   3. $\sqrt{gL\cos\theta}$
   4. $\sqrt{2gL\cos\theta}$
6. A mass $m$ is attached to a spring and oscillates on a frictionless horizontal surface. What is the total energy of the system in terms of the amplitude $A$ and spring constant $k$?
   1. $\frac{1}{4}kA^2$
   2. $\frac{1}{2}kA$
   3. $\frac{1}{2}kA^2$
   4. $kA^2$
7. A car of mass $m$ moving with velocity $v$ applies brakes and comes to rest after covering a distance $d$. If the frictional force is $f$, what is the work done by friction?
   1. $fd$
   2. $-fd$
   3. $\frac{1}{2}mv^2$
   4. $-\frac{1}{2}mv^2$