### Problem Statement**84.** Boxes A and B in Figure P10.84 have masses of 12.0 kg and 4.0 kg, respectively. The two boxes are released from rest. Use conservation of energy to find the boxes' speed when box B has fallen a distance of 0.50 m. Assume a frictionless upper surface.### Diagram Description**Figure P10.84** shows a horizontal surface on which Box A (mass of 12.0 kg) rests. A pulley system connects Box A to Box B (mass of 4.0 kg) with a rope. Box B hangs vertically off the edge of the surface. When released, Box B falls a distance of 0.50 meters vertically downward, while Box A moves horizontally. The pulley and the surface beneath Box A are assumed to be frictionless.### Solution and ExplanationThe setup involves the use of conservation of mechanical energy to determine the speed of the boxes. Since there is no friction, we only need to consider the conversion of potential energy to kinetic energy.1. **Initial Energy:** - Potential Energy (U) of Box B before it falls: \[ U_{\text{initial}} = m_B \cdot g \cdot h \] where: \[ m_B = 4.0 \, \text{kg}, \quad g = 9.8 \, \text{m/s}^2, \quad h = 0.50 \, \text{m} \] Substituting the values: \[ U_{\text{initial}} = 4.0 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 \cdot 0.50 \, \text{m} = 19.6 \, \text{J} \]2. **Final Energy:** - When Box B has fallen 0.50 m, it loses its gravitational potential energy, which gets converted into the kinetic energy of both boxes. - The kinetic energy (K) of Box A and Box B: \[ K_{\text{final}} = \frac{1}{2} m_A v^2 + \frac{1}{2} m_B v^2 \] Simplifying: \[ K_{\

Given: The mass of box A is 12 kg. The mass of the block B is 4 kg. The initial speed of the boxes…

84. Boxes A and B in Figure P10.84 have masses of 12.0 kg and 4.0 kg, respectively. The two boxes are released from rest. Use conservation of energy to find the boxes' speed when box B has fallen a distance of 0.50 m. Assume a frictionless upper surface. A B

84. Boxes A and B in Figure P10.84 have masses of 12.0 kg and 4.0 kg, respectively. The two boxes are released from rest. Use conservation of energy to find the boxes' speed when box B has fallen a distance of 0.50 m. Assume a frictionless upper surface. A B

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Question

### Problem Statement

**84.** Boxes A and B in Figure P10.84 have masses of 12.0 kg and 4.0 kg, respectively. The two boxes are released from rest. Use conservation of energy to find the boxes' speed when box B has fallen a distance of 0.50 m. Assume a frictionless upper surface.

### Diagram Description

**Figure P10.84** shows a horizontal surface on which Box A (mass of 12.0 kg) rests. A pulley system connects Box A to Box B (mass of 4.0 kg) with a rope. Box B hangs vertically off the edge of the surface. When released, Box B falls a distance of 0.50 meters vertically downward, while Box A moves horizontally. The pulley and the surface beneath Box A are assumed to be frictionless.

### Solution and Explanation

The setup involves the use of conservation of mechanical energy to determine the speed of the boxes. Since there is no friction, we only need to consider the conversion of potential energy to kinetic energy.

1. **Initial Energy:**
- Potential Energy (U) of Box B before it falls:
\[
U_{\text{initial}} = m_B \cdot g \cdot h
\]
where:
\[
m_B = 4.0 \, \text{kg}, \quad g = 9.8 \, \text{m/s}^2, \quad h = 0.50 \, \text{m}
\]
Substituting the values:
\[
U_{\text{initial}} = 4.0 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 \cdot 0.50 \, \text{m} = 19.6 \, \text{J}
\]

2. **Final Energy:**
- When Box B has fallen 0.50 m, it loses its gravitational potential energy, which gets converted into the kinetic energy of both boxes.
- The kinetic energy (K) of Box A and Box B:
\[
K_{\text{final}} = \frac{1}{2} m_A v^2 + \frac{1}{2} m_B v^2
\]
Simplifying:
\[
K_{\

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