Problem 49 - College Physics

Question

A machine part consists of a thin, uniform \(4.00-\mathrm{kg}\) bar that is 1.50 \(\mathrm{m}\) long, hinged perpendicular to a similar vertical bar of mass 3.00 \(\mathrm{kg}\) and length 1.80 \(\mathrm{m} .\) The longer bar has a small but dense 2.00 -kg ball at one end (Fig. 8.42\()\) By what distance will the center of mass of this part move horizontally and vertically if the vertical bar is pivoted counterclockwise through \(90^{\circ}\) to make the entire part horizontal?

Step-by-Step Solution

Answer

Horizontal movement: 0.56 m right; Vertical movement: 0.25 m down.

Step 1: Understand the System

The system consists of two bars hinged together - a horizontal bar and a vertical bar. The horizontal bar has a concentrated mass at one end.

Step 2: Determine Initial Center of Mass Horizontal Position

Initially, only the 1.50 m bar is horizontal, so calculate the contributions of horizontal positions to the system's center of mass. The masses are placed at the bar's pivot and the ball’s endpoint.

Step 3: Calculate Initial Center of Mass Vertical Position

Calculate the vertical position of the center of mass. Initially, for the vertical bar only, this is halfway along its length, and for the horizontal bar, at its pivot.

Step 4: Determine Final Center of Mass Horizontal Position

After the 90 degree rotation, calculate the new horizontal positions for each mass. The vertical bar is now horizontal, so its mass and that of the ball need new x-coordinates.

Step 5: Calculate Final Center of Mass Vertical Position

Calculate the new vertical position for each mass. The vertical bar will now be entirely aligned horizontally, so it contributes zero to the vertical coordinate. The horizontal bar stays unchanged in position.

Step 6: Calculate Change in Horizontal Position

Subtract the initial horizontal center of mass from the final horizontal center of mass to find the change in horizontal distance.

Step 7: Calculate Change in Vertical Position

Subtract the initial vertical center of mass from the final vertical center of mass to find the change in vertical distance.

Key Concepts

Physics Problem Solving

When it comes to solving complex problems in physics, having a structured approach is crucial. In this exercise, the task is to determine the movement of the center of mass of a machine part when one of its components pivots. To break this down:

• First, understand the system. Visualize the configuration of the machine part, including each bar and mass point.
• Next, identify all known parameters, such as lengths, masses, and pivot points.
• Once the system's components are clear, calculate the initial positions using the given dimensions and masses.
• Afterward, simulate the effect of the pivoting by recalculating these positions.

Each of these steps builds on the previous one, allowing you to systematically solve the problem. This structured approach is a cornerstone of effective problem-solving in physics.

Mechanics

In mechanics, understanding motion is essential, including how forces affect the behavior of objects. In exercises like this, we need to consider:

• The properties of each object, such as mass distribution and shape, which determine the initial and final positions of the center of mass.
• How each part of the system, such as a bar or a ball, contributes to the overall dynamics.
• The relation between static configurations and dynamic transformations, like rotation around a pivot point.

By focusing on these key aspects, mechanics helps us predict movements and understand how shifts in one part of a system affect the whole. For this problem, we're interested in seeing how a once vertical rod contributes to new positions when it's moved horizontally.

Rotational Motion

Rotational motion deals with objects rotating around a point or axis. For this particular problem, we focus on how the vertical bar pivots to change the system's configuration. Key points include:

• The concept of pivoting, which involves rotating around a fixed point without sliding.
• Understanding how each point mass's new position affects the overall center of mass calculation.
• Recognizing that the rotation effectively shifts the contributions of position from a vertical to a horizontal orientation.

By mastering these rotational dynamics, you gain insight into how every pivot affects distances and angles. It's essential for visualizing how the center of mass moves when a body rotates.

Static Equilibrium

Static equilibrium refers to a state where all forces in a system are balanced, and there is no resultant motion. In solving exercises that involve pivoting like this one:

• Initially, we consider each part of the system in a static position before and after rotation.
• We analyze how the center of mass changes, ensuring that the system remains balanced relative to the new orientation.
• The calculation of center of mass before and after the movement illustrates how it maintains equilibrium through shifts in position.

The concept of static equilibrium helps us understand balance within mechanical systems, maintaining structural integrity while translating forces within the pivot's context.