Question

A 0.500 -kg mass is attached to a horizontal spring with \(k=100 . \mathrm{N} / \mathrm{m}\) The mass slides across a frictionless surface. The spring is stretched \(25.0 \mathrm{~cm}\) from equilibrium, and then the mass is released from rest. a) Find the mechanical energy of the system. b) Find the speed of the mass when it has moved \(5.00 \mathrm{~cm}\). c) Find the maximum speed of the mass.

Short answer

Mechanical energy: 3.125 J Speed of the mass when it has moved 5 cm: 1.50 m/s Maximum speed of the mass: 2.50 m/s

Step-by-step solution

a) Find the mechanical energy of the system.

At the initial position, the mass is at rest, and the spring is stretched by 25 cm (0.25 m). To find the mechanical energy, we need to find the potential energy stored in the spring. The potential energy in a spring is given by the formula: \(U_{spring} = \frac{1}{2}kx^2\) where \(U_{spring}\) is the potential energy, \(k\) is the spring constant, and \(x\) is the displacement from the equilibrium position. Plugging in the given values, we get: \(U_{spring} = \frac{1}{2}(100\,\text{N/m})(0.25\,\text{m})^2\) \(U_{spring} = 3.125\,\text{J}\) Since there is no kinetic energy at this point (the mass is at rest), the total mechanical energy of the system is equal to the potential energy stored in the spring, which is 3.125 J. So the mechanical energy of the system is: \(E_{mech} = 3.125\,\text{J}\)

b) Find the speed of the mass when it has moved 5.00 cm.

When the mass has moved 5 cm (0.05 m), the spring is now compressed by 0.2 m (0.25 - 0.05) compared to its initial stretched position. We can find the potential energy at this point: \(U_{spring} = \frac{1}{2}kx^2 = \frac{1}{2}(100\,\text{N/m})(0.2\,\text{m})^2 = 2.0\,\text{J}\) By conservation of mechanical energy, the kinetic energy at this point is the difference between the total mechanical energy and the potential energy: \(K = E_{mech} - U_{spring} = 3.125\,\text{J} - 2.0\,\text{J} = 1.125\,\text{J}\) Now, we can use the kinetic energy formula to find the speed of the mass: \(K = \frac{1}{2}mv^2\) \(v = \sqrt{\frac{2K}{m}}\) \(v = \sqrt{\frac{2(1.125\,\text{J})}{(0.5\,\text{kg})}}\) \(v = 1.50\,\text{m/s}\) So the speed of the mass when it has moved 5 cm is 1.50 m/s.

c) Find the maximum speed of the mass.

The maximum speed will occur when the potential energy of the spring is 0, which happens when the spring returns to its equilibrium position. At this point, the total mechanical energy of the system is equal to the kinetic energy: \(K = E_{mech} = 3.125\,\text{J}\) Using the kinetic energy formula, we can find the maximum speed: \(K = \frac{1}{2}mv^2\) \(v_{max} = \sqrt{\frac{2K}{m}}\) \(v_{max} = \sqrt{\frac{2(3.125\,\text{J})}{(0.5\,\text{kg})}}\) \(v_{max} = 2.50\,\text{m/s}\) So the maximum speed of the mass is 2.50 m/s.

Key concepts

Mechanical Energy Conservation

Understanding mechanical energy conservation is essential when analyzing the behavior of a mass-spring system. Mechanical energy, in this context, consists of two major types: potential energy stored in the spring (also known as elastic potential energy), and kinetic energy, which is the energy of motion. According to the conservation of mechanical energy principle, the total mechanical energy in a frictionless system remains constant. This means that when a mass attached to a spring moves, its kinetic energy and potential energy can interchange, but their sum does not change.

For example, at the point of maximum stretch or compression of the spring, the kinetic energy is zero, and all the mechanical energy is stored as potential energy. As the mass moves towards equilibrium, potential energy is converted into kinetic energy. The equation for mechanical energy conservation looks like this:
\[E_{mech} = U_{spring} + K\]
Where \(E_{mech}\) is the total mechanical energy, \(U_{spring}\) is the potential energy in the spring, and \(K\) is the kinetic energy of the mass. By making use of this key concept, we can easily solve problems concerning the motion of masses attached to springs, as shown in the step-by-step solution provided.

Simple Harmonic Motion

Simple harmonic motion (SHM) describes a type of periodic motion or oscillation motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. A good example of SHM is the motion of a mass attached to a spring when it is displaced from its equilibrium position and then released.

The spring's restoring force follows Hooke's Law, which states that the force exerted by a spring is proportional to the displacement:\[F = -kx\]
Here, \(F\) is the force, \(k\) is the spring constant, and \(x\) is the displacement of the mass from its equilibrium position. The negative sign indicates that the force acts in the opposite direction of the displacement.

In the context of the given exercise, SHM is demonstrated as the mass-spring system oscillates back and forth about the equilibrium position. This motion can be analyzed using the principles of mechanical energy conservation and is characterized by its amplitude, frequency, and period, providing rich insights into the dynamical behavior of oscillatory systems.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It is formally defined by the equation:
\[K = \frac{1}{2}mv^2\]
Where \(K\) represents kinetic energy, \(m\) is the mass of the object, and \(v\) is its velocity. In the scenario presented in the exercise, kinetic energy becomes a significant variable when the mass attached to the spring starts moving from its stretched position towards the equilibrium point.

As the mass moves and speed increases, the potential energy stored in the spring converts into kinetic energy. The maximum kinetic energy is achieved when the spring passes through the equilibrium position where the potential energy is momentarily zero. As we calculate the kinetic energy of the mass at different points, we apply the conservation of mechanical energy to determine its velocity, demonstrating the interchange between potential and kinetic forms of energy within a frictionless system.