3. A spring of constant k, compressed a distance x, is used to launch a mass m up a frictionless slope at angle 0. Find an expression for the maximum distance along the slope that the mass moves after leaving the spring.

Physics for Scientists and Engineers
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ISBN:9781337553278
Author:Raymond A. Serway, John W. Jewett
Publisher:Raymond A. Serway, John W. Jewett
Chapter7: Energy Of A System
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**Problem 3: Analysis of a Spring-Launched Mass on a Frictionless Incline**

A spring with a constant \( k \) is compressed by a distance \( x \) and is used to propel a mass \( m \) up an incline that is completely frictionless and angled at \( \theta \). Determine an expression for the maximum distance the mass will travel along the incline after it detaches from the spring.

**Explanation:**

To solve this problem, we need to analyze the energy transformations involved:

1. **Initial Energy**: The initial energy comes from the potential energy stored in the compressed spring, given by \( \frac{1}{2} k x^2 \).

2. **Final Energy**: As the mass moves up the incline, the spring’s potential energy transforms into gravitational potential energy. At the maximum distance, the kinetic energy will be zero because the mass momentarily stops.

3. **Gravitational Potential Energy Increase**: As the mass moves up, it's potential energy changes by \( mgh \), where \( h \) can be expressed in terms of the distance up the incline \( d \), using \( h = d \sin \theta \).

**Energy Conservation Equation**:
\[ \frac{1}{2} k x^2 = mg(d \sin \theta) \]

**Solving for \( d \)**:
\[ d = \frac{\frac{1}{2} k x^2}{mg \sin \theta} \]

The above equation provides the expression for the maximum distance \( d \) the mass travels along the incline.
Transcribed Image Text:**Problem 3: Analysis of a Spring-Launched Mass on a Frictionless Incline** A spring with a constant \( k \) is compressed by a distance \( x \) and is used to propel a mass \( m \) up an incline that is completely frictionless and angled at \( \theta \). Determine an expression for the maximum distance the mass will travel along the incline after it detaches from the spring. **Explanation:** To solve this problem, we need to analyze the energy transformations involved: 1. **Initial Energy**: The initial energy comes from the potential energy stored in the compressed spring, given by \( \frac{1}{2} k x^2 \). 2. **Final Energy**: As the mass moves up the incline, the spring’s potential energy transforms into gravitational potential energy. At the maximum distance, the kinetic energy will be zero because the mass momentarily stops. 3. **Gravitational Potential Energy Increase**: As the mass moves up, it's potential energy changes by \( mgh \), where \( h \) can be expressed in terms of the distance up the incline \( d \), using \( h = d \sin \theta \). **Energy Conservation Equation**: \[ \frac{1}{2} k x^2 = mg(d \sin \theta) \] **Solving for \( d \)**: \[ d = \frac{\frac{1}{2} k x^2}{mg \sin \theta} \] The above equation provides the expression for the maximum distance \( d \) the mass travels along the incline.
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