Question

use conservation of energy to determine the unknown energies.

Explanation:

Step1: Recall energy - conservation principle

The total mechanical energy \(E = GPE+KE +\text{Losses}\) (where \(GPE\) is gravitational - potential energy, \(KE\) is kinetic energy) is conserved in the absence of non - conservative forces other than the ones accounted for. At point \(a\), since \(E_{int}=0\) and there are no losses and no work done externally, the total mechanical energy \(E_a\) is just the kinetic energy. So \(E_a = KE_a\).

Step2: Calculate total energy at point \(b\)

At point \(b\), \(E_b=GPE_b + KE_b+\text{Losses}_b\). Given \(GPE_b = 330J\), \(KE_b = 90J\), and \(\text{Losses}_b=0J\), \(E_b=330 + 90=420J\). Also, work done \(W = 715J\) is added to the system.

Step3: Calculate energy at point \(c\)

The total energy at point \(c\) is \(E_c=E_b + W-\text{Losses}\). Given \(E_b = 420J\), \(W = 715J\), and \(\text{Losses}=100J\), \(E_c=420+715 - 100=1035J\). At point \(c\), \(E_c=GPE_c+KE_c + E_{int}\). Given \(E_{int}=320J\) and \(KE_c = 320J\), we can find \(GPE_c\). Since \(E_c = 1035J\), \(GPE_c=1035-(320 + 320)=395J\).

Answer:

The unknown gravitational - potential energy at point \(c\) is \(395J\).