Work:

Work: the product of the force and the distance travelled in the direction of the force

$$ W=\vec{F}\cdot \vec{x} $$

Case study:

• An object moves in a straight line with a varying force $F(x)$ then the Work becomes the area under the curve:

$$ W=\int^{x_1}_{x_0}F(x)\,\text{d}x $$

• An object moves along a curved path with a varying force $F(r)$ the work becomes:

$$ W=\int^{r_2}{r_1}\vec{F}\cdot\text{d}\vec{r}=\int^{r_2}{r_1}F(r)\cos{\theta}\,\text{d}r $$

Work Energy Theorem:

Definition: The work done by the resultant force on a particle is equal to the change in the kinetic energy of the particle $W=\Delta E_k$

Case study:

• An object of mass $m$ has a constant resultant force $F$ acting on it moving along a straight line in the $x$-axis such that the speed goes from $v_0$ to $v$ during the displacement $x$:$\begin{aligned}

\end{aligned}$

$$ \begin{aligned} v^2=v^2_0+2ax &\Leftrightarrow a=\frac{v^2-v^2_0}{2x} \;\text{ The force is:}\\ F=ma &\Rightarrow F=m\frac{v^2-v^2_0}{2x} \;\text{ The work is:}\\ W=Fx \Rightarrow W=m\frac{v^2-v^2_0}{2x}x &\Leftrightarrow W=\frac{1}{2}mv^2-\frac{1}{2}mv^2_0 \Rightarrow W=\Delta E_k \end{aligned} $$

• An object of mass $m$ has a varying resultant force $F$ acting on it, moving along a straight line in the $x$-axis:

$$ \begin{aligned} W=\int \vec{F}\cdot\text{d}\vec{r} &= \int^x_{x_0}F(x)\text{d}x \\ \text{using: } F=ma=m\frac{\text{d}v}{\text{d}t}\text{ a}&\text{nd: }a=\frac{\text{d}v}{\text{d}t}=\frac{\text{d}v}{\text{d}x}\frac{\text{d}x}{\text{d}t}=v\frac{\text{d}v}{\text{d}x} \\ W=\int^x_{x_0}F\text{d}x=\int^x_{x_0}ma\,\text{d}x&=\int^x_{x_0}mv\frac{\text{d}v}{\text{d}x}\text{d}x \\ \frac{\text{d} v}{\text{d} x} \text{d} x=\text{d} v \Rightarrow W =\int_{v_0}^v m v\, \text{d} v&=\left[\frac{mv^2}{2}\right]_{v_0}^v=\frac{1}{2}mv^2-\frac{1}{2}m v_0^2 \Rightarrow W=\Delta E_k \end{aligned} $$

• An object of mass $m$ has a varying resultant force F acting on it, moving along a curved path:

$$ \begin{aligned} \vec{F}=m\frac{\text{d}\vec{v}}{\text{d}t} \text{ and } \text{d}\vec{r}=\vec{v}\text{d}t &\Rightarrow W=\int^a_b\vec{F}\cdot\text{d}\vec{r}=\int^{t_a}{t_b}m\frac{\text{d}\vec{v}}{\text{d}t}\cdot\vec{v}\text{d}t \\ \frac{\text{d}\vec{v}}{\text{d}t}\cdot\vec{v}=\frac{1}{2}\frac{\text{d}(v^2)}{\text{d}t} &\Rightarrow W=\int{t_a}^{t_b} \frac{1}{2}m \frac{\text{d} (v^2)}{\text{d} t}\text{d} t=\frac{1}{2} mv_b^2-\frac{1}{2}mv_a^2\Rightarrow W=\Delta E_k \end{aligned}
$$

Power:

Definition: Power is the rate of doing work