6. You are designing a pulley/counterweight system to lift fuel rods in and out of a nuclear reactor. The fuel rods have mass of 2M and the counterweight has a mass of 1.5M. The mass and friction of the pulley are negligible. The rods are initially at rest how fast are they moving after 2M block has dropped a distance H? a. Determine the speed using conservation of mechanical energy. 1.5M 2M b. Repeat the problem using Newton's 2nd law and constant acceleration kinematics. Show that you get the same answer as in (a).

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**Problem 6: Pulley and Counterweight System Design**

You are designing a pulley/counterweight system to lift fuel rods in and out of a nuclear reactor. The fuel rods have a mass of \(2M\) and the counterweight has a mass of \(1.5M\). The mass and friction of the pulley are negligible. The rods are initially at rest. How fast are they moving after the \(2M\) block has dropped a distance \(H\)?

**a. Determine the speed using conservation of mechanical energy.**

**b. Repeat the problem using Newton’s 2nd law and constant acceleration kinematics. Show that you get the same answer as in (a).**

**Diagram Explanation:**

The diagram illustrates a simple pulley system. The pulley is attached to the ceiling. On the left side of the rope, a mass labeled \(1.5M\) represents the counterweight, and on the right side, a mass labeled \(2M\) represents the fuel rods. The weight of each side is supported by the tension in the rope, and the acceleration due to gravity is represented as \(g\). The pulley is assumed to be frictionless and massless, simplifying the analysis.
Transcribed Image Text:**Problem 6: Pulley and Counterweight System Design** You are designing a pulley/counterweight system to lift fuel rods in and out of a nuclear reactor. The fuel rods have a mass of \(2M\) and the counterweight has a mass of \(1.5M\). The mass and friction of the pulley are negligible. The rods are initially at rest. How fast are they moving after the \(2M\) block has dropped a distance \(H\)? **a. Determine the speed using conservation of mechanical energy.** **b. Repeat the problem using Newton’s 2nd law and constant acceleration kinematics. Show that you get the same answer as in (a).** **Diagram Explanation:** The diagram illustrates a simple pulley system. The pulley is attached to the ceiling. On the left side of the rope, a mass labeled \(1.5M\) represents the counterweight, and on the right side, a mass labeled \(2M\) represents the fuel rods. The weight of each side is supported by the tension in the rope, and the acceleration due to gravity is represented as \(g\). The pulley is assumed to be frictionless and massless, simplifying the analysis.
Expert Solution
Step 1

a)

Let speed of both is v after moving by H

By energy conservation 

(1/2)(1.5M+2M)v= 2MgH - 1.5gH 

3.5v= gH 

v = √(2gH/7) 

b)  by Newton's law 

By fbd of system 

acceleration (a) = (2Mg-1.5Mg)/(2M+1.5M) 

a =  g/7 

Now by equation of motion

v2 = u2 +2as 

v2 = 0 + 2(g/7)H 

v = √(2gH/7)

 

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