If the loss in graviational potential energy to falling the sphere by h height and heat loss to surrounding at constant rate `H` are also taken to account the energy equation will modify to
(A) `m_(1) s_(1) (theta_(1)-theta) + (m_(1)gh)/(J) = m_(2) s_(2) (theta - theta_(2)) + m_(3) s_(3) (theta - theta_(2)) - H t`
(B) `m_(1)s_(1) (theta_(1) -theta) - (m_(1)gh)/(J) = m_(2)s_(2)(theta -theta_(2)) + m_(3) s_(3) (theta -theta_(2)) + Ht`
(C) `m_(1)s_(1) (theta_(1) -theta) + (m_(1)gh)/(J) = m_(2) s_(2) (theta - theta_(2)) + m_(3) s_(3) (theta -theta_(2)) + Ht`
(D) `m_(1) s_(1) (theta_(1)-theta)-(m_(1)gh)/(J)=m_(2)s_(2)(theta-theta_(2)) +m_(3)s_(3)(theta-theta_(2))-Ht` .

Explanation

To solve the problem, we need to analyze the energy equation considering both the loss in gravitational potential energy when a sphere falls by a height \( h \) and the heat loss to the surroundings at a constant rate \( H \).

### Step-by-Step Solution:

1. **Understanding Gravitational Potential Energy (GPE)**:
The gravitational potential energy lost when the sphere falls by a height \( h \) is given by:
\[
\text{GPE} = m_1 g h
...