Question

A \(5.13-\mathrm{kg}\) object moves on a horizontal frictionless surface under the influence of a spring with force constant \(9.88 \mathrm{~N} / \mathrm{cm}\). The object is displaced \(53.5 \mathrm{~cm}\) and given an initial velocity of \(11.2 \mathrm{~m} / \mathrm{s}\) back toward the equilibrium position. Find \((a)\) the frequency of the motion, \((b)\) the initial potential energy of the system, \((c)\) the initial kinetic energy, and \((d)\) the amplitude of the motion.

Short answer

The frequency of the motion is approximately 1.39 Hz, the initial potential energy is 14 J, the initial kinetic energy is 32.1 J, and the amplitude of the motion is 0.535 m.

Step-by-step solution

Calculating Frequency of Motion

The frequency of motion of a spring-object system is given by \(f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}\), where k is the spring constant and m is the mass of the object. First convert the spring constant from N/cm to N/m (1 cm = 0.01 m), so \( k = 9.88 N/cm * 0.01 m/cm = 98.8 N/m\). Now we plug the values \(k = 98.8 N/m\) and \(m = 5.13 kg\) into the equation and solve for frequency.

Calculate Initial Potential Energy of the system

The initial potential energy of a spring system when it's compressed or stretched is given by \(PE = \frac{1}{2} k x^2\), where x is the displacement of the object. To find the initial potential energy, we substitute the values \(x = 0.535 m\) (converted from cm to m) and \(k = 98.8 N/m\) into the equation and solve it.

Calculate Initial Kinetic Energy of the system

The kinetic energy of an object in motion is given by equation \(KE = \frac{1}{2} m v^2\). To find the kinetic energy, the values \(m = 5.13 kg\) and \(v = 11.2 m/s\) are replaced into the equation and the expression is solved.

Find the Amplitude of Motion

In simple harmonic motion, the amplitude is the maximum displacement from equilibrium. Since the initial velocity is towards the equilibrium, the initial displacement \(x = 0.535 m\) is the amplitude of the motion.

Key concepts

Frequency of Motion

The frequency of motion in simple harmonic motion (SHM) describes how often an object oscillates back and forth over a given time period. In the context of an oscillating spring-object system, frequency signifies the number of complete cycles the object makes per second. Using the formula
\(f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}\),
where \(f\) is frequency, \(k\) is the spring constant, and \(m\) is the object's mass, we can calculate how quickly the system oscillates. For an object attached to a spring, as in our exercise, the frequency depends on the stiffness of the spring (the spring constant \(k\)) and the mass of the object (\(m\)). It's important to note that the unit of spring constant should be in newtons per meter (N/m), and mass in kilograms (kg) to apply this formula correctly.

Potential Energy

Potential energy in SHM is the stored energy when the spring is compressed or stretched from its equilibrium position. It is highest when the object is at its maximum displacement, where all the energy of the system is momentarily stored as potential energy. The formula to calculate this energy is
\(PE = \frac{1}{2} k x^2\),
where \(PE\) stands for potential energy, \(k\) for the spring constant, and \(x\) for the displacement from the equilibrium position. As we compress or stretch the spring further from the equilibrium position, the potential energy increases. Hence, the displacement \(x\) has a crucial role in determining the potential energy of the system.

Kinetic Energy

Kinetic energy is the energy of motion. In simple harmonic motion, kinetic energy is at its maximum when the object passes through the equilibrium position. This is because the object travels the fastest at this point. Using the equation
\(KE = \frac{1}{2} m v^2\),
we calculate kinetic energy, where \(m\) represents the mass of the object and \(v\) its velocity. In the step-by-step solution, the initial kinetic energy is found when the object is given an initial push. The velocity during SHM changes constantly, and thus, so does kinetic energy. It's an essential concept to understand as it directly relates to the object's speed and mass.

Amplitude of Motion

Amplitude of motion is the maximum distance the object moves from its equilibrium position during simple harmonic motion. It is a crucial measurement as it defines the limits of the oscillation and is a key in determining the total energy of the system. The amplitude is not dependant on mass or spring constant but is determined by the initial conditions of the system, like how far it is pulled or pushed from equilibrium and then released. In the given exercise, the amplitude is the initial displacement provided, which signifies how much the spring was initially stretched.

Spring Constant

The spring constant, symbolized by \(k\), quantifies the stiffness of a spring. Technically, it measures the force required to compress or stretch the spring by a unit length. The spring constant is central to Hooke's Law, which is foundational in understanding elastic potential energy in springs. The greater the spring constant, the stiffer the spring, and the more force is needed to stretch or compress it by the same amount. This stiffness influences how the spring returns to its equilibrium position and thus affects the frequency of the oscillating system as shown in our exercise.

Equilibrium Position

In simple harmonic motion, the equilibrium position is where the object would naturally rest if there were no net forces acting upon it. It is a point at which the forces due to the spring (restoring force) and any other external forces balance out. When the object is displaced either side of this point and released, it oscillates about this equilibrium position. This position is significant when examining the motion dynamics since the object experiences the maximum restoring force, and consequently the maximum acceleration, when it is furthest from the equilibrium.