Problem 83 - University Physics with Modern Physics

Question

A 1.00 -kg mass is suspended vertically from a spring with \(k=100 . \mathrm{N} / \mathrm{m}\) and oscillates with an amplitude of \(0.200 \mathrm{~m} .\) At the top of its oscillation, the mass is hit in such a way that it instantaneously moves down with a speed of \(1.00 \mathrm{~m} / \mathrm{s}\). Determine a) its total mechanical energy, b) how fast it is moving as it crosses the equilibrium point, and c) its new amplitude.

Step-by-Step Solution

Answer

Answer: The total mechanical energy is 2.50 J, the speed at the equilibrium point is approximately 2.24 m/s, and the new amplitude is approximately 0.224 m.

Step 1: Calculate the initial conditions

First, let's find the initial kinetic energy (K.E) and potential energy (P.E) of the mass at the top of its oscillation. The initial kinetic energy is given by the formula: \(K.E = \cfrac{1}{2}mv^2\), where \(m\) is the mass, and \(v\) is the initial velocity. We're given the mass \(m=1.00 \text{ kg}\) and the initial velocity \(v=1.00 \text{ m/s}\). So, \(K.E = \cfrac{1}{2}(1.00)(1.00^2) = 0.50 \text{ J}\) (Joules) Now let's find the initial potential energy due to the spring. The potential energy is given by the formula: \(P.E = \cfrac{1}{2}kx^2\), where \(k\) is the spring constant and \(x\) is the displacement (amplitude). We're given the spring constant \(k = 100 \text{ N/m}\) and the amplitude \(x=0.200 \text{ m}\). So, \(P.E = \cfrac{1}{2}(100)(0.200^2) = 2.00 \text{ J}\)

Step 2: Calculate the total mechanical energy

Total mechanical energy (E) is the sum of kinetic energy and potential energy. So, \(E = K.E + P.E = 0.50 \text{ J} + 2.00 \text{ J} = 2.50 \text{ J}\) Therefore, the total mechanical energy of the mass is \(2.50 \text{ J}\).

Step 3: Find the speed at the equilibrium point

At the equilibrium point, the potential energy is zero, so the total mechanical energy of the system is equal to the kinetic energy. Using the formula for kinetic energy \(K.E = \cfrac{1}{2}mv^2\) and our total mechanical energy, we can solve for the velocity at the equilibrium point: \(2.50 \text{ J} = \cfrac{1}{2}(1.00 \text{ kg})v^2\) \(v^2 = \cfrac{2.50*2}{1.00} \Rightarrow v = \sqrt{5.00}\) \(v \approx 2.24 \text{ m/s}\) So, the speed as it crosses the equilibrium point is approximately \(2.24 \text{ m/s}\).

Step 4: Find the new amplitude

To find the new amplitude, we use the conservation of mechanical energy. At the new maximum height (amplitude), the kinetic energy will be zero, and the total mechanical energy will be equal to the potential energy due to the spring: \(E = \cfrac{1}{2}kx^2\). We know the total mechanical energy (E) and the spring constant (k), so we can solve for the new amplitude (x): \(2.50 \text{ J} = \cfrac{1}{2}(100) x^2\) \(x^2 = \cfrac{2.50*2}{100} \Rightarrow x = \sqrt{0.0500}\) \(x \approx 0.224 \text{ m}\) So, the new amplitude of oscillation is approximately \(0.224 \text{ m}\). To summarize: a) The total mechanical energy is \(2.50 \text{ J}\). b) The speed as it crosses the equilibrium point is approximately \(2.24 \text{ m/s}\). c) The new amplitude is approximately \(0.224 \text{ m}\).

Key Concepts

Spring Constant

The spring constant, symbolized by the letter 'k', is a measure of the stiffness of a spring. In the context of the problem, the spring constant applies to Hooke's Law, which states that the force exerted by the spring is proportional to the displacement from its equilibrium position. Mathematically, we express this as

\( F = kx \),

where \( F \) is the force, \( k \) is the spring constant, and \( x \) is the displacement. A high spring constant indicates a stiffer spring that requires more force to stretch or compress by a given amount. In the given exercise, understanding the spring constant is crucial as it allows us to calculate the potential energy stored in the spring when the mass is suspended.

Kinetic Energy

Kinetic energy represents the energy that an object possesses due to its motion. For an object with mass \( m \) and velocity \( v \), the kinetic energy is given by the formula

\( K.E = \frac{1}{2}mv^2 \).

In the provided exercise, the kinetic energy of the mass changes as it moves up and down, reaching its maximum as the mass passes through the equilibrium point. It's important to connect this concept with the motion described: initially, the mass has kinetic energy imparted by the hit, and as it oscillates, this energy dynamically converts into potential energy and back.

Potential Energy

Potential energy is the energy stored within a system as a result of its position or configuration. For a mass attached to a spring, the potential energy, denoted P.E, is a function of the displacement from the equilibrium and the spring constant. The formula for the potential energy in a spring system is

\( P.E = \frac{1}{2}kx^2 \),

where \( k \) is the spring constant and \( x \) is the displacement from equilibrium. In our exercise, we use this formula to calculate the potential energy at the top of the oscillation. Potential energy is at its maximum at the amplitude of oscillation, and it is zero at the equilibrium position, indicating a transfer of energy types throughout the motion.

Amplitude of Oscillation

The amplitude of oscillation is the maximum displacement of an object from its rest position. In harmonic motion, amplitude represents the peak distance the mass moves from the equilibrium position and is directly related to the energy in the system. The initial amplitude in the problem is given, and then we calculate a new one after the mass is struck. It's key to understand that the amplitude is an indicator of the maximum potential energy in the system, as when the object is at the amplitude, all its mechanical energy is potential.

Conservation of Mechanical Energy

The conservation of mechanical energy principle states that the total mechanical energy in a system remains constant if only conservative forces (like gravity and spring forces) are doing work. The total mechanical energy is the sum of kinetic energy and potential energy. In equations, we express this as

\( E = K.E + P.E \).

When applying this concept to the exercise, we utilize the fact that at the top and bottom of the oscillation, the energy is purely potential, while at the equilibrium point, it is purely kinetic. By knowing the total mechanical energy, we can calculate various parameters of the system at different points in the oscillation.

Harmonic Motion

Harmonic motion, or simple harmonic motion, is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. This oscillatory motion is characteristic of pendulums, springs, and other systems that can oscillate freely. In our exercise, understanding harmonic motion is vital to predicting how the mass will behave when it's disturbed from its initial conditions and how its kinetic and potential energies change over time.