A ball of unknown mass m is tossed straight up with initial speed u. At the moment it is released, the ball is a height h above a spring-mounted platform, as shown in the figure. The ball rises, peaks, and falls back toward the platform, ultimately compressing the spring a maximum distance d 5 from its relaxed position. Assume that the spring is perfectly ideal, with spring constant k, and that the mass of the spring and platform is negligible. What is the mass of the ball, m, assuming that there is no friction or air resistance? Using g to represent the acceleration due to gravity, enter an expression for m in terms of g, h, d, k, and v. m h

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**Title: Calculating Mass Using Energy Principles**

**Problem Statement:**

A ball of unknown mass \( m \) is tossed straight up with an initial speed \( v \). At the moment it is released, the ball is a height \( h \) above a spring-mounted platform, as shown in the figure. The ball rises, peaks, and falls back toward the platform, ultimately compressing the spring a maximum distance \( d \) from its relaxed position. Assume that the spring is perfectly ideal, with spring constant \( k \), and that the mass of the spring and platform is negligible.

**Objective:**

What is the mass of the ball, \( m \), assuming that there is no friction or air resistance?

Using \( g \) to represent the acceleration due to gravity, enter an expression for \( m \) in terms of \( g, h, d, k, \) and \( v \).

**Solution:**

\[
m = \frac{\frac{1}{2} k d^2}{gh}
\]

**Diagram Explanation:**

- *Left Diagram:* The ball is shown just above the spring-mounted platform at height \( h \). The ball is moving upward with initial speed \( v \).

- *Right Diagram:* The spring is compressed by a distance \( d \), indicating that the ball has fallen and compressed the spring to its maximum displacement.

This setup helps illustrate energy transformation: the initial kinetic and potential energy is converted to elastic potential energy in the compressed spring. The solution involves equating these energy transformations to solve for the ball's mass \( m \).
Transcribed Image Text:**Title: Calculating Mass Using Energy Principles** **Problem Statement:** A ball of unknown mass \( m \) is tossed straight up with an initial speed \( v \). At the moment it is released, the ball is a height \( h \) above a spring-mounted platform, as shown in the figure. The ball rises, peaks, and falls back toward the platform, ultimately compressing the spring a maximum distance \( d \) from its relaxed position. Assume that the spring is perfectly ideal, with spring constant \( k \), and that the mass of the spring and platform is negligible. **Objective:** What is the mass of the ball, \( m \), assuming that there is no friction or air resistance? Using \( g \) to represent the acceleration due to gravity, enter an expression for \( m \) in terms of \( g, h, d, k, \) and \( v \). **Solution:** \[ m = \frac{\frac{1}{2} k d^2}{gh} \] **Diagram Explanation:** - *Left Diagram:* The ball is shown just above the spring-mounted platform at height \( h \). The ball is moving upward with initial speed \( v \). - *Right Diagram:* The spring is compressed by a distance \( d \), indicating that the ball has fallen and compressed the spring to its maximum displacement. This setup helps illustrate energy transformation: the initial kinetic and potential energy is converted to elastic potential energy in the compressed spring. The solution involves equating these energy transformations to solve for the ball's mass \( m \).
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