Question

directions: solve the problem shown below. any data, calculations, or models you use to support your thinking should be explained. problem: a vertical spring is attached to the floor, as shown in figure 1. an object of mass ( m_1 ) is placed on top of the spring and slowly lowered, compressing the spring a distance ( x_1 ) from its unstretched length, until it reaches equilibrium, as shown in figure 2. next, an object of mass ( m_2 ) is placed on top of the first object and slowly lowered, compressing the spring an additional distance ( x_2 ). the system is again in equilibrium in figure 3. what is the relationship between ( x_2 ) and ( x_1 )?

Explanation:

Step1: Analyze equilibrium for \( m_1 \)

When only \( m_1 \) is on the spring, at equilibrium, the spring force equals the weight of \( m_1 \). Using Hooke's Law \( F = kx \) (where \( k \) is the spring constant and \( x \) is the compression) and Newton's second law (net force is zero), we have:
\( kx_1 = m_1g \) (1)

Step2: Analyze equilibrium for \( m_1 + m_2 \)

When \( m_2 \) is added on top of \( m_1 \), the total compression is \( x_1 + x_2 \). At equilibrium, the spring force equals the total weight of \( m_1 \) and \( m_2 \):
\( k(x_1 + x_2) = (m_1 + m_2)g \) (2)

Step3: Solve for \( x_2 \) in terms of \( x_1 \)

Expand equation (2): \( kx_1 + kx_2 = m_1g + m_2g \). From equation (1), we know \( kx_1 = m_1g \). Substitute \( kx_1 \) in equation (2) with \( m_1g \):
\( m_1g + kx_2 = m_1g + m_2g \).
Subtract \( m_1g \) from both sides: \( kx_2 = m_2g \).

Now, from equation (1), \( k = \frac{m_1g}{x_1} \). Substitute \( k \) into \( kx_2 = m_2g \):
\( \frac{m_1g}{x_1} \cdot x_2 = m_2g \).
Cancel \( g \) from both sides: \( \frac{m_1x_2}{x_1} = m_2 \).
Rearrange to solve for \( x_2 \): \( x_2 = \frac{m_2}{m_1}x_1 \).

(Note: If we assume \( m_2 = m_1 \) (though not stated, a common case for proportionality), \( x_2 = x_1 \). But generally, \( x_2 \) is proportional to \( x_1 \) with the ratio \( \frac{m_2}{m_1} \). However, if the problem implies \( m_2 = m_1 \) (e.g., equal masses), then \( x_2 = x_1 \).)

Answer:

If \( m_2 = m_1 \), then \( \boldsymbol{x_2 = x_1} \). More generally, \( \boldsymbol{x_2 = \frac{m_2}{m_1}x_1} \) (proportional to \( x_1 \) with ratio \( \frac{m_2}{m_1} \)).