4.2 Work and Mechanical Energy

Key Concepts in Work and Mechanical Energy

Changes in Kinetic Energy

Kinetic energy ($KE$) represents the energy of motion. It depends on an object’s mass and velocity:

$$KE=\frac{1}{2}m{v}^2$$

Where:

• $m$: Mass of the object (kg).
• $v$: Velocity of the object (m/s).

Key Points:

• Proportional Relationship: Kinetic energy increases with the square of velocity. Doubling velocity quadruples $KE$.
• External Forces: Changes in $KE$ occur when forces like friction, gravity, or collisions act on an object.
• Formula for Change in Kinetic Energy:

$$\mathrm{\Delta }KE=K{E}_{\text{final}}-K{E}_{\text{initial<span style="font-family: Georgia, 'Times New Roman', 'Bitstream Charter', Times, serif; font-size: 16px;"></span>}}$$

Changes in Total Energy

Interactions with external objects or systems can transfer energy, altering a system’s total energy.

• Work: Transfers energy to or from a system when a force acts over a displacement.
• Mechanical Energy: Sum of $KE$ and potential energy ($PE$).

Forces and Work

The work done on an object depends on:

1. The force applied.
2. The displacement of the object.
3. The angle between the force and displacement vectors.

Work is calculated as:

$$W=Fd\cos \theta$$

Where:

• $F$: Force (N).
• $d$: Displacement (m).
• $\theta$: Angle between force and displacement.

Types of Energy

Kinetic Energy ($KE$)

Work can increase $KE$, demonstrated by the work-energy principle:

$$W=\mathrm{\Delta }KE$$

Derivation:

For an object of mass $m$ acted upon by a net force $F$ over a displacement $d$:

1. Use Newton’s Second Law: $F = ma$.
2. Use kinematic equations to relate acceleration and displacement.
3. Substitute into the work formula to derive $\Delta KE$.

Gravitational Potential Energy ($PE_g$)

Work done to change an object’s position relative to a reference point increases its gravitational potential energy:

$$P{E}_g=mgh$$

Where:

• $m$: Mass of the object (kg).
• $g$: Gravitational acceleration ($9.8 \, \text{m/s}^2$).
• $h$: Height above the reference point (m).

In gravitational systems beyond Earth, $PE_g$ can also be calculated using:

$$P{E}_g=-\frac{G{m}_1{m}_2}{\mathrm{r<span\; class="vlist"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"><span\; class="msupsub"><span\; class="vlist-s"></span></span></span><span\; class="vlist-s"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"></span>}}$$

Where $r$ is the distance between the two masses’ centers.

Elastic Potential Energy ($PE_e$)

Elastic potential energy is stored in deformable objects like springs. For an ideal spring:

$$P{E}_e=\frac{1}{2}k{x}^2$$

Where:

• $k$: Spring constant (N/m).
• $x$: Displacement from equilibrium (m).

The spring constant can be determined from a plot of force ($F_s$) vs. displacement ($x$), where the area under the curve represents the work done.

Thermal Energy

Thermal energy is a catch-all for non-mechanical energy, such as heat or sound:

• Often arises from friction or collisions.
• In AP Physics 1, no formal equation is required, but it’s a vital part of energy conservation in systems.

Mechanical Energy Conservation

For a closed system with no non-conservative forces:

$$K{E}_{\text{initial}}+P{E}_{\text{initial}}=K{E}_{\text{final}}+P{E}_{\text{final<span style="font-family: Georgia, 'Times New Roman', 'Bitstream Charter', Times, serif; font-size: 16px;"></span>}}$$

Key Takeaways on Energy Types

1. Kinetic Energy: Motion energy proportional to mass and velocity squared.
2. Potential Energy:
   • Gravitational: Depends on height and mass.
   • Elastic: Stored in springs or deformable materials.
3. Thermal Energy: Energy dissipated as heat or sound.
4. Mechanical Energy: Sum of $KE$ and $PE$.

Real-World Applications

1. Roller Coasters:

   • Convert $PE_g$ at the top of a hill into $KE$ as the coaster descends.

2. Springs:

   • Elastic potential energy powers systems like trampolines or shock absorbers.

3. Heat from Friction:

   • Tires heating up as they skid due to friction.

Practice Problem

Problem:

A 2 kg object moves at 3 m/s. How much work is required to increase its velocity to 6 m/s?

Solution:

1. Calculate initial $KE$:

$$KE_{\text{initial}} = \frac{1}{2}(2)(3)^2 = 9 \, \text{J}$$

2. Calculate final $KE$:

$$KE_{\text{final}} = \frac{1}{2}(2)(6)^2 = 36 \, \text{J}$$

3. Work done ($W$):

$$W = \Delta KE = KE_{\text{final}} – KE_{\text{initial}} = 36 – 9 = 27 \, \text{J}$$