Problem 23 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A block of mass \(m\) is attached to the end of a spring (spring stiffness constant \(k\) ), Fig. 8-35. The mass is given an initial displacement \(x_{0}\) from equilibrium, and an initial speed \(v_{0}\). Ignoring friction and the mass of the spring, use energy methods to find \((a)\) its maximum speed, and \((b)\) its maximum stretch from equilibrium, in terms of the given quantities.

Step-by-Step Solution

Answer

(a) Maximum speed: \( v_{max} = \sqrt{v_0^2 + \frac{k}{m} x_0^2} \); (b) Maximum stretch: \( x_{max} = \sqrt{x_0^2 + \frac{m}{k} v_0^2} \)."

Step 1: Understanding Energy Conservation

We begin by considering the conservation of mechanical energy. Since there is no friction, the total mechanical energy in the system is conserved. The mechanical energy when the spring is stretched by an initial displacement \( x_0 \) and given an initial speed \( v_0 \) consists of kinetic and potential energy. Let's write this initial energy as \( E_i = \frac{1}{2} m v_0^2 + \frac{1}{2} k x_0^2 \).

Step 2: Calculating Maximum Speed (a)

The maximum speed of the block occurs when all of the potential energy is converted into kinetic energy. At this point, the spring is at its equilibrium position \( x = 0 \). Thus, the energy at maximum speed \( E_k = \frac{1}{2} m v_{max}^2 \) equals the initial energy: \( \frac{1}{2} m v_{max}^2 = \frac{1}{2} m v_0^2 + \frac{1}{2} k x_0^2 \). Solving for \( v_{max} \) gives: \[ v_{max} = \sqrt{v_0^2 + \frac{k}{m} x_0^2} \].

Step 3: Determining Maximum Stretch (b)

The maximum stretch occurs when all kinetic energy is converted into potential energy at the maximum displacement \( x_{max} \). At this point, \( E_p = \frac{1}{2} k x_{max}^2 \) is equal to the initial energy: \( \frac{1}{2} k x_{max}^2 = \frac{1}{2} m v_0^2 + \frac{1}{2} k x_0^2 \). Solving for \( x_{max} \) involves rearranging to get: \[ x_{max} = \sqrt{x_0^2 + \frac{m}{k} v_0^2} \].

Key Concepts

Harmonic Motion

In physics, harmonic motion refers to the type of motion that repeats itself in a regular cycle. It is most commonly associated with objects that oscillate, such as a mass attached to a spring. This type of motion is termed "simple harmonic motion" because it is characterized by the restoring force that is directly proportional to the displacement and acts in the opposite direction. The force can be described by Hooke's Law, given by \( F = -kx \), where \( F \) is the restoring force, \( k \) is the spring stiffness constant, and \( x \) is the displacement from equilibrium.
In the context of the exercise, the block on the spring undergoes harmonic motion as it moves back and forth around its equilibrium position. The displacement, speed, and acceleration of the block change in a predictable way over time, following a sinusoidal pattern. While examining these factors, it's crucial to understand how energy conservation plays a role throughout the cycle, leading to periods of maximum speed and maximum stretch.

Potential Energy

Potential energy is an essential concept in understanding the energy within a system in physics. It is the energy stored in an object due to its position or arrangement. For a spring-mass system, potential energy is specifically the energy stored in the spring when it is compressed or stretched from its equilibrium position.

• The potential energy, stored in the spring, depends on the spring constant \( k \), a measure of the spring's stiffness, and the displacement \( x \) from its equilibrium position. This relationship is given by the formula \( PE = \frac{1}{2} k x^2 \).
• In the exercise, potential energy is initially present when the spring is either compressed or extended by displacement \( x_0 \). At any point during the oscillation of the block, the potential energy adjusts according to the displacement from equilibrium.

The conservation of mechanical energy indicates that as potential energy increases (e.g., at maximum stretch), kinetic energy decreases, and vice versa. This transfer back and forth allows the system to maintain a constant total mechanical energy when no external forces, such as friction, are acting on it.

Kinetic Energy

Kinetic energy, in contrast to potential energy, refers to the energy an object possesses due to its motion. In the context of our spring-mass system, the kinetic energy depends on both the mass of the block \( m \) and its velocity \( v \). The formula used to calculate kinetic energy is given by \( KE = \frac{1}{2} m v^2 \).

• Initially, when the mass is displaced and set in motion, it possesses some kinetic energy due to its velocity \( v_0 \).
• As the block oscillates, its kinetic energy fluctuates, reaching a maximum when the spring is at its equilibrium (unstrained position) and its velocity is highest.

According to the problem, the maximum kinetic energy occurs when the block moves through its equilibrium position. At this point, all the energy forms present in the system have fully converted to kinetic energy. This is illustrated by reaching the calculated maximum speed \( v_{max} \). Understanding how kinetic energy relates to the harmonic motion helps provide a vivid picture of how the block's motion and energy relationship play out, seamlessly transitioning between maximum kinetic and potential energy states.