Final Answer:
The acceleration ar(t) of the rod is expressed as ar(t) = (-b/m) * vr(t), where m represents the mass of the rod, vr(t) is the velocity of the rod, and b signifies a constant related to resistance or friction.
Step-by-step explanation:
The acceleration of the rod, ar(t), can be understood through Newton's second law, F = ma, where F denotes the net force acting on the rod, m is the mass, and a is the acceleration. Considering the resistance force proportional to the velocity of the rod, we apply the equation of motion for resistance, F = -bv. Rearranging to isolate acceleration, a = F/m = (-b/m) * v.
In this context, the resistance force opposing the motion of the rod is proportional to its velocity, represented by -bv, where 'b' is a constant related to the resisting medium (like air or friction) and 'v' is the velocity of the rod. By dividing this force by the mass 'm,' we derive the acceleration. Therefore, the acceleration ar(t) = (-b/m) * vr(t) shows that the rate of change of velocity of the rod is inversely proportional to its mass ('m') and directly proportional to the resisting factor 'b' and the velocity of the rod, vr(t).
This equation highlights that as the mass of the rod increases, the acceleration decreases, while an increase in the resistance constant 'b' or the velocity of the rod leads to a higher acceleration opposing its motion. It elucidates the dynamic relationship between mass, resistance, and velocity in determining the acceleration of the rod.