Unit 3 – Work, Energy, and Power

3.1 Translational Kinetic Energy

Kinetic energy refers to the energy of an object’s motion.

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

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Claim: “if you double a block’s speed, its kinetic energy will double”

$$K_A=\frac{1}{2}m{v}^2$$ $$K_B=\frac{1}{2}m(2v{)}^2$$ $$K_B=4\cdot K_A$$

The claim is incorrect. The block will have four times the kinetic energy.

3.2 Work

Work is mechanical energy transferred into or out of a system. Work is the force times the distance.

$$W=F\cdot d\cdot \cos \theta$$

$F$ is the force exerted on the system, $d$ is the distance over with the force is exerted, $\theta$ is the angle between the force and the distance.

The area under a Force vs Distance graph is the work.

Work is equal to the change in kinetic energy.

$$W=\Delta K$$

3.3 Potential Energy

Elastic Potential Energy

Elastic potential energy is potential energy stored in an ideal spring.

$$U_s=\frac{1}{2}k(\Delta x{)}^2$$

$U_s$ is the scalar amount of jules. $k(N/m)$ is the spring constant. $\Delta x(\mathrm{m})$ is the distance the spring has been stretched or compressed from its equilibrium length.

$$U_1=\frac{1}{2}k(0{)}^2$$ $$U_1=0$$ $$U_2=\frac{1}{2}(x{)}^2$$ $$U_3=\frac{1}{2}k(2x{)}^2$$ $$U_3=4U_2$$

Local Gravitational Potential Energy

Gravitational potential energy ($U_g$) is the potential energy caused by an object’s change in position in a gravitational field.

$$U_g=mg\Delta y$$

Total Gravitational Potential Energy

Similar to the Gravitational Force equation, the potential energy of objects very far from earth have a different equation.

$$U_g=-\frac{G{m}_1{m}_2}{r}$$

3.4 Conservation of Energy

The sum of $K$, $U_g$, and $U_s$ will remain the same in a closed system.

How fast is the car traveling at $x$?

$$K=\frac{1}{2}m{v}^2$$ $$U_g=mg\Delta y$$ $$\frac{1}{2}m{v}^2=mg\Delta y$$ $$v=\sqrt{2g\Delta y}$$ $$v=\sqrt{2\cdot 9.8\cdot 15}$$ $$v=17.146m/s$$

How fast is the car traveling at $y$?

$$K=\frac{1}{2}m{v}^2$$ $$U_g=mg\Delta y$$ $$\frac{1}{2}m{v}^2=mg\Delta y$$ $$v=\sqrt{2g\Delta y}$$ $$v=\sqrt{2\cdot 9.8\cdot 5}$$ $$v=9.899m/s$$

Practice Question

A $5.0\mathrm{kg}$ block is attached to a spring as shown below and released from rest. At point $A$, the block leaves the spring and slides over a rough surface before coming to a stop at point $B$.

What is the speed of the block at point A?

$$\frac{1}{2}k(\Delta x{)}^2=\frac{1}{2}m{v}^2$$ $$v=\sqrt{\frac{k(\Delta x{)}^2}{m}}$$ $$v=4.21m/s$$

What is the distance between point A and point B?

$$W=F\cdot d$$ $$\frac{1}{2}m{v}^2=(\mu \cdot mg)d$$ $$d=\frac{m{v}^2}{2\mu \cdot mg}$$ $$d=\frac{v^2}{2\mu \cdot g}$$ $$d=\frac{4.21^2}{2\cdot 0.35\cdot 9.8}$$ $$d=2.584\mathrm{m}$$

3.5 Power

Work is the amount of mechanical energy transferred into or out of a system. Power is the rate at which work is done or the rate at which energy changes.

$$P_{\text{avg}}=\frac{W}{\Delta t}=\frac{\Delta E}{\Delta t}$$

Unit: watt ($W$)

Example Question

A student collects the following data, determine how much power was used in the first travel to travel up the stairs.

Trial 1
Mass of person	$55\mathrm{kg}$
Time	$22\mathrm{s}$
Number of stairs	$14$
Step height	$17\mathrm{cm}$

$$P=\frac{mg\Delta y}{22}$$ $$P=\frac{55\cdot 9.8\cdot 14\cdot 0.17}{22}$$ $$P=58.31\mathrm{W}$$