D3 and D5 answer ASAP a rigid rod is moving with velocity u in vertical direction

When analyzing the motion of a rigid rod moving vertically with a velocity \( u \), we can consider several important aspects, including the forces acting on the rod, its center of mass, and the implications of its motion in a gravitational field. Let’s break this down step by step.

Understanding the Motion of the Rod

A rigid rod can be thought of as a straight line segment with a fixed length, and it can rotate about its center of mass or any other point. When this rod is moving vertically with a constant velocity \( u \), we need to consider the following:

• Direction of Motion: The rod is moving straight up or down, depending on the sign of \( u \). If \( u \) is positive, it moves upward; if negative, downward.
• Forces Acting on the Rod: The primary force acting on the rod is gravity, which pulls it downward with a force equal to its weight (mass \( m \) times gravitational acceleration \( g \)).
• Center of Mass: The center of mass of the rod will also move with the same velocity \( u \), as long as there are no other forces acting on it horizontally.

Equations of Motion

In this scenario, we can apply the basic equations of motion. Since the rod is moving vertically, we can use the following kinematic equation to describe its position over time:

\( y(t) = y_0 + ut + \frac{1}{2}gt^2 \)

Here, \( y(t) \) is the vertical position of the rod at time \( t \), \( y_0 \) is the initial position, and \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)). Note that if the rod is moving upward, \( g \) will act in the opposite direction, effectively reducing the upward velocity over time.

Effects of Gravity

As the rod moves upward, gravity will decelerate its motion. If the rod is initially moving upward, it will eventually reach a maximum height where its velocity becomes zero before starting to fall back down. The time taken to reach this maximum height can be calculated using:

\( t_{max} = \frac{u}{g} \)

After reaching the maximum height, the rod will begin to descend, accelerating downwards due to gravity.

Example Scenario

Let’s consider a specific example. Imagine a rigid rod of length 2 meters and mass 5 kg is moving upward with a velocity of 10 m/s. The forces acting on it are:

• Weight of the rod: \( W = mg = 5 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 49.05 \, \text{N} \) downward.
• Initial upward velocity: \( u = 10 \, \text{m/s} \).

Using the equations of motion, we can find the maximum height the rod will reach before it starts to fall back down. The time to reach maximum height is:

\( t_{max} = \frac{10 \, \text{m/s}}{9.81 \, \text{m/s}^2} \approx 1.02 \, \text{s} \)

During this time, the rod will ascend to a height calculated by substituting \( t_{max} \) into the position equation:

\( y(t_{max}) = y_0 + 10 \times 1.02 - \frac{1}{2} \times 9.81 \times (1.02)^2 \approx y_0 + 10.2 - 5.1 \approx y_0 + 5.1 \, \text{m} \)

This example illustrates how the motion of a rigid rod in a vertical direction can be analyzed using basic principles of physics, considering both its initial velocity and the effects of gravity. Understanding these concepts is crucial for solving more complex problems in dynamics.