We will analyze the motion of the system using Newton’s second law and kinematic equations.

Step 1: Identify Forces in the System

• The force applied at point B is $F = 10$ lb.
• Block A has a weight of $W_A = 18$ lb.
• The acceleration due to gravity is $g = 32.2 \text{ ft/s}^2$.

Since the pulleys and cords are massless, we assume an ideal system where tension is uniformly distributed.

Step 2: Determine Acceleration of Block A

Assume the pulleys are arranged in such a way that pulling the cord at B down a distance $s_B = 3$ ft causes Block A to move up a corresponding distance $s_A$.

Using Newton’s Second Law for Block A: $\sum F = m_A a_A$

The mass of Block A is: $m_A = \frac{W_A}{g} = \frac{18}{32.2} \approx 0.559 \text{ slugs}$

Since the force at B is being used to lift Block A, we analyze the system considering the effective tension:

If the pulley system distributes force equally, then the tension in the cord is: $T = 10 \text{ lb}$

The net force on Block A is: $T - W_A = m_A a_A$

Substituting values: $10 - 18 = (0.559) a_A$

$-8 = 0.559 a_A$

$a_A = \frac{-8}{0.559} \approx -14.3 \text{ ft/s}^2$

Since the acceleration is negative, Block A moves downward, meaning the assumption about directions must be corrected.

Step 3: Kinematics to Find Time

Using the kinematic equation: $s = v_0 t + \frac{1}{2} a t^2$

Since the motion starts from rest ($v_0 = 0$): $s = \frac{1}{2} a t^2$

For $s_B = 3$ ft and using the magnitude of acceleration: $3 = \frac{1}{2} (14.3) t^2$

$3 = 7.15 t^2$

$t^2 = \frac{3}{7.15} \approx 0.42$

$t \approx 0.65 \text{ s}$

Final Answer:

The time required to pull the cord at B down 3 ft is 0.65 seconds.

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Follow-Up Questions

1. How does the number of pulleys affect the acceleration of Block A?
2. What happens if the applied force at B is increased to 15 lb?
3. How would adding friction to the pulleys impact the motion?
4. What would happen if Block A had a different weight?
5. How does this problem change if the system starts with an initial velocity?

Tip:

Always check the direction of acceleration when applying Newton’s Second Law, especially in pulley problems.