Problem 69 - University Physics with Modern Physics

Question

A \(0.150-\mathrm{kg}\) block of ice is placed against a horizontal, compressed spring mounted on a horizontal tabletop that is 1.20 \(\mathrm{m}\) above the floor. The spring has force constant 1900 \(\mathrm{N} / \mathrm{m}\) and is initially compressed 0.045 \(\mathrm{m}\) . The mass of the spring is negligible. The spring is released, and the block slides along the table, goes off the edge, and travels to the floor. If there is negligible friction between the block of ice and the tabletop, what is the speed of the block of ice when it reaches the floor?

Step-by-Step Solution

Answer

The speed of the block when it hits the floor is approximately 5.98 m/s.

Step 1: Determine Potential Energy in the Spring

The potential energy stored in the spring when compressed is given by the formula: \( PE_{spring} = \frac{1}{2} k x^2 \), where \( k \) is the spring constant and \( x \) is the compression distance. Here, \( k = 1900 \, \mathrm{N/m} \) and \( x = 0.045 \, \mathrm{m} \). Calculate this energy as follows:\[ PE_{spring} = \frac{1}{2} \cdot 1900 \, \mathrm{N/m} \cdot (0.045 \, \mathrm{m})^2 = 1.924125 \, \mathrm{J} \]

Step 2: Convert Spring Energy to Kinetic Energy

When the spring is released, its potential energy is fully converted to kinetic energy at the moment the ice block leaves the spring. According to the conservation of energy, \( KE = PE_{spring} \). Hence, the kinetic energy of the block is:\[ KE = 1.924125 \, \mathrm{J} \]

Step 3: Determine Speed on Table

The kinetic energy \( KE \) can be expressed in terms of the speed \( v \) of the block as \( KE = \frac{1}{2} mv^2 \). Solve for \( v \):\[ \frac{1}{2} \cdot 0.150 \, \mathrm{kg} \cdot v^2 = 1.924125 \, \mathrm{J} \]\[ v^2 = \frac{2 \cdot 1.924125}{0.150} \]\[ v^2 = 25.655 \rightarrow v = \sqrt{25.655} \approx 5.065 \, \mathrm{m/s} \]

Step 4: Calculate Speed When Reaching the Floor

When the block leaves the table, it has initial horizontal speed \( v_x = 5.065 \, \mathrm{m/s} \). It falls a vertical distance of \( 1.20 \, \mathrm{m} \), gaining speed from gravitational acceleration. Using the energy conservation, sum of kinetic energies:\[ \frac{1}{2}mv^2 + mgh = \frac{1}{2}mv_f^2 \]Here, \( v_f \) is final speed, \( h = 1.20 \, \mathrm{m} \), and \( g = 9.81 \, \mathrm{m/s^2} \):\[ 1.924125 + 0.150 \cdot 9.81 \cdot 1.20 = \frac{1}{2}\cdot 0.150 \cdot v_f^2 \]\[ v_f = \sqrt{10.71} \approx 3.27 \, \mathrm{m/s} \]

Step 5: Calculate Combined Speed Upon Impacting Floor

The block retains its horizontal speed \( v_x = 5.065 \, \mathrm{m/s} \) and its vertical speed becomes \( 3.27 \, \mathrm{m/s} \). The total speed \( v \) can be found using Pythagorean Theorem:\[ v = \sqrt{v_x^2 + v_y^2} \]\[ v = \sqrt{(5.065)^2 + (3.27)^2} \approx \sqrt{35.86} \approx 5.98 \, \mathrm{m/s} \]

Key Concepts

Potential Energy

Potential energy is a form of stored energy. In this exercise, it refers to the energy stored in the compressed spring. When a spring is compressed or stretched, it stores energy that can be used later. To calculate spring potential energy, we use the formula:

• \( PE_{spring} = \frac{1}{2}kx^2 \)

Here, \( k \) is the spring constant, and \( x \) is the displacement from the spring's equilibrium position. In our example, the spring was compressed by 0.045 meters, and had a spring constant of 1900 N/m. Thus, the potential energy stored in the spring was approximately 1.92 Joules. This stored energy is what propels the block of ice from rest when the spring is released.

Spring Constant

The spring constant \( k \) is a measure of a spring's stiffness. It is crucial in determining how much force is needed to compress or stretch a spring by a certain distance. A higher spring constant means a stiffer spring, which requires more force for the same displacement.In our exercise, the spring constant was given as 1900 N/m. This indicates a relatively stiff spring. Knowing the spring constant is essential because it directly affects how much potential energy the spring can store when compressed. Using this constant along with the compression distance, we can determine the potential energy using the formula provided earlier.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. Once the potential energy in the spring is released, it is converted entirely into kinetic energy, causing the block of ice to move. This illustrates the principle of conservation of energy, where energy changes forms but is not lost.The formula for kinetic energy is:

• \( KE = \frac{1}{2}mv^2 \)

where \( m \) is the mass of the object and \( v \) is its velocity. In this context, the initial kinetic energy of the block after leaving the spring was 1.92 Joules, calculated using the block's mass and initial velocity. As such, this tells us how fast the block moves immediately after the spring propels it.

Gravitational Acceleration

Gravitational acceleration is the acceleration of an object due to the force of gravity. On Earth, this acceleration is approximately \( 9.81 \, \text{m/s}^2 \). As the block of ice slides off the table and falls, it gains speed because of gravitational acceleration.The change in the vertical velocity of the block when it falls from the table height of 1.20 meters is calculated by considering the energy conversion from potential energy to kinetic energy:

• Potential energy at height: \( mgh \)
• \( h = 1.20 \, \text{m} \) and \( g = 9.81 \, \text{m/s}^2 \)

Using these values, we find the block gains further speed due to gravity as it falls, resulting in a vertical speed that combines with the horizontal to find the total speed upon hitting the ground.