Question

At time ${\mathit{t}}_{\mathbf{i}}$, the kinetic energy of a particle is $\mathbf{30}\mathbf{.}\mathbf{0}\mathbf{}\mathit{J}$and the potential energy of the system to which it belongs is $\mathbf{10}\mathbf{.}\mathbf{0}\mathbf{}\mathit{J}$. At some later time, the kinetic energy of the particle is $\mathbf{18}\mathbf{.}\mathbf{0}\mathbf{}\mathit{J}$. (a) If only conservative forces act on the particle, what are the potential energy and the total energy of the system at time ${\mathit{t}}_{\mathbf{f}}$? (b) If the potential energy of the system at time ${\mathit{t}}_{\mathbf{f}}$ is $\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{}\mathit{J}$, are any non conservative forces acting on the particle? (c) Explain your answer to part (b).

Short answer

(a) The total energy of the system at time $t_f$is$E=40J$

(b) Yes; if the potential energy is less than $22J$

(c) The total mechanical energy has decreased, so a non conservative force must have acted.

Step-by-step solution

Definition of the energy

Consider the book moves on an incline that is not frictionless, there is a change in both the kinetic energy and the gravitational potential energy of the book Earth system. Consequently, $\Delta E_{mech}=\Delta K+\Delta U_g=-f_{k}d=-\Delta E_{int}$

In general, if a non conservative force acts within an isolated system

$\sum W_{otherforces}=\Delta K+\Delta U+\Delta E_{int}$

where

$\Delta k=$Change in kinetic energy

$\Delta U=$Change in potential energy

$\Delta E_{int}=$ Change in internal energy.

Calculating the total energy of the system

(a)

If only conservative forces act, then the total mechanical energy does not change.

By using the concept from step (1), write the conservation of energy equation for this situation.

$$\begin{gathered} \sum W_{otherforces}=\Delta K+\Delta U+\Delta E_{int} \\ \Delta K+\Delta U=0 \end{gathered}$$

Substitute the initial and final energies.

$$\begin{gathered} \left(k_f-k_i\right)+\left(u_f-u_i\right)=0 \\ {u}_f=k_f-k_f+u_i \end{gathered}$$

Substitute numerical values.

$$\begin{gathered} u_f=30J-18J+10J \\ {u}_f=22J \end{gathered}$$

Now

$$\begin{gathered} E=K+U \\ E=30J+10J \\ E=40J \end{gathered}$$

Comparing the potential energy

(b)

Yes, if the potential energy is less than $22J$$22J$

Explaining the reason for b

(c)

If the potential energy is$5.00J$, the total mechanical energy is

$$\begin{gathered} E=K+U \\ E=18J+5J=23J \end{gathered}$$

Less than the original$40J$.

The total mechanical energy has decreased, so a non-conservative force must have acted.