Problem 66 - College Physics

Question

Two identical 1.50 \(\mathrm{kg}\) masses are pressed against opposite ends of a light spring of force constant \(1.75 \mathrm{N} / \mathrm{cm},\) compress- ing the spring by 20.0 \(\mathrm{cm}\) from its normal length. Find the speed of each mass when it has moved free of the spring on a frictionless horizontal lab table.

Step-by-Step Solution

Answer

Speeds of both masses are approximately 1.53 m/s.

Step 1: Understand the Problem

Two identical masses are pressed against the ends of a spring, compressing it by a certain distance. We need to find the speed of each mass when the spring returns to its normal length. Since the table surface is frictionless and there is no external force, the system is isolated. This indicates that the mechanical energy is conserved.

Step 2: Apply Conservation of Energy

The mechanical energy at the initial state is stored as potential energy in the spring since the masses are at rest. The potential energy (PE) stored in the spring is given by:\[PE = \frac{1}{2} k x^2\]where \(k = 1.75 \mathrm{N/cm} = 175 \mathrm{N/m}\) (converted) and \(x = 0.2 \mathrm{m}\) (in meters). The final energy when the spring is fully relaxed is purely kinetic as the potential energy will be zero.

Step 3: Calculate Initial Potential Energy

Substitute the values into the potential energy formula to find the initial potential energy:\[PE = \frac{1}{2} \times 175 \times (0.2)^2 = 3.5 \mathrm{J}\]This value represents the total initial energy of the system.

Step 4: Equate Potential Energy to Kinetic Energy

When each mass leaves the spring, all initial potential energy is converted into kinetic energy. Since there are two masses, the kinetic energy of each mass is:\[2(KE) = 3.5 \mathrm{J}\]where \(KE = \frac{1}{2} mv^2\) for each mass (both masses share the potential energy equally).

Step 5: Solve for Speed of Each Mass

Calculate the kinetic energy for one mass:\[KE = \frac{3.5}{2} = 1.75 \mathrm{J}\]Then, using the kinetic energy formula:\[1.75 = \frac{1}{2} \times 1.5 \times v^2\]Solving for \(v\):\[v^2 = \frac{1.75 \times 2}{1.5}\]\[v^2 = \frac{3.5}{1.5} = 2.333\]\[v = \sqrt{2.333} \approx 1.53 \mathrm{m/s}\]

Step 6: Conclude the Solution

The speed of each mass when it moves free of the spring on the frictionless table is approximately \(1.53 \mathrm{m/s}\). This result is derived from conservation of energy and assuming the system is isolated.

Key Concepts

Potential Energy

Potential energy is a type of energy that is stored due to an object's position or state. In our exercise, the potential energy is stored in a compressed spring. The formula to calculate potential energy in a spring is given by \[ PE = \frac{1}{2} k x^2 \] where

• \(k\) represents the spring constant, measured in Newtons per meter (N/m), which is a measure of the spring's stiffness.
• \(x\) is the displacement from the spring's equilibrium position—in this case, it's how much the spring has been compressed or stretched, measured in meters.

When the spring in the problem is compressed, it stores energy that can be released to do work on the masses. This stored potential energy is pivotal in determining the masses' motion when the spring returns to its normal length.

Kinetic Energy

Kinetic energy is the energy of motion. When the spring in this exercise releases its potential energy, it is converted into the kinetic energy of the moving masses. The kinetic energy of an object is given by the formula:\[ KE = \frac{1}{2} mv^2 \] where

• \(m\) is the mass of the object, in kilograms.
• \(v\) represents the velocity or speed of the object, in meters per second.

In a frictionless environment, like the one described in the problem, all the potential energy stored in the spring becomes kinetic energy when the spring releases, causing the masses to move. The faster the masses move, the more kinetic energy they possess, derived from the total original potential energy.

Mechanical Energy

Mechanical energy is simply the sum of potential energy and kinetic energy within a system. In this problem, the conservation of mechanical energy principle is crucial. Initially, the total mechanical energy is entirely in the form of potential energy stored in the compressed spring. As the spring returns to its normal length, this potential energy transforms into kinetic energy of the masses: \[ \text{Mechanical Energy} = \text{Potential Energy} + \text{Kinetic Energy} \] Due to the conservation of energy, the total mechanical energy in an isolated system remains constant. The energy initially stored in the spring equals the total kinetic energy of the masses after leaving the spring.

Frictionless Surface

A frictionless surface is an idealized concept where there is no resistance to the motion of objects. In this scenario, the frictionless horizontal table provides an environment where energy losses that usually occur due to friction are eliminated. This makes it easier to apply the conservation of mechanical energy principle.

• No friction means that when the masses are released from the spring, they convert all stored potential energy into kinetic energy without any loss.
• A real-world non-frictionless surface would convert some energy into thermal energy due to friction, reducing the kinetic energy available to the masses.

In this exercise, the frictionless setup simplifies calculations and completion of the solution by ensuring energy conservation is straightforward and exact.