Astronauts in space cannot weigh themselves by standing on a bathroom scale. Instead, they determine their mass by oscillating on a large spring. Suppose an astronaut attaches one end of a large spring to her belt and the other end to a hook on the wall of the space capsule. A fellow astronaut then pulls her away from the wall and releases her. The spring's length as a function of time is shown in FIGURE P15.46. What is her speed when the spring's length is 1.2 m?

Verified step by step guidance

1. Step 1: Analyze the graph provided. The graph shows the spring's length (L) as a function of time (t). The oscillation is periodic, with a maximum length of 1.8 m and a minimum length of 0.6 m. The period of oscillation is 4 seconds, as the pattern repeats every 4 seconds.
2. Step 2: Determine the equilibrium position of the spring. The equilibrium length is the average of the maximum and minimum lengths: \( L_{eq} = \frac{L_{max} + L_{min}}{2} \). Substitute \( L_{max} = 1.8 \ \text{m} \) and \( L_{min} = 0.6 \ \text{m} \) into the formula.
3. Step 3: Use the conservation of energy principle to relate the potential energy stored in the spring and the kinetic energy of the astronaut. The total mechanical energy in the system is constant and is given by \( E = \frac{1}{2} k A^2 \), where \( k \) is the spring constant and \( A \) is the amplitude of oscillation. The speed of the astronaut can be found using \( v = \sqrt{\frac{k}{m} (A^2 - (L - L_{eq})^2)} \), where \( m \) is the astronaut's mass, \( L \) is the spring's length at the given moment, and \( L_{eq} \) is the equilibrium length.
4. Step 4: Identify the amplitude \( A \) of oscillation from the graph. The amplitude is the difference between the maximum length and the equilibrium length: \( A = L_{max} - L_{eq} \). Substitute the values of \( L_{max} \) and \( L_{eq} \) to find \( A \).
5. Step 5: Substitute the given spring length \( L = 1.2 \ \text{m} \), the calculated equilibrium length \( L_{eq} \), the amplitude \( A \), and the spring constant \( k \) into the velocity formula \( v = \sqrt{\frac{k}{m} (A^2 - (L - L_{eq})^2)} \). This will give the astronaut's speed at the specified spring length.

Key Concepts

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. In this case, the astronaut oscillates due to the restoring force of the spring, which is proportional to the displacement from the equilibrium position. The motion is characterized by a sinusoidal pattern, which can be described mathematically by sine or cosine functions.

Spring Constant (k)

The spring constant, denoted as 'k', is a measure of a spring's stiffness. It is defined by Hooke's Law, which states that the force exerted by a spring is proportional to its displacement from the equilibrium position (F = -kx). A higher spring constant indicates a stiffer spring, which affects the frequency and amplitude of the oscillation experienced by the astronaut.

Velocity in Oscillatory Motion

In oscillatory motion, the velocity of an object varies with time and is maximum at the equilibrium position and zero at the maximum displacement. The velocity can be derived from the displacement function of the oscillation, which is often sinusoidal. To find the speed at a specific displacement, such as when the spring's length is 1.2 m, one can differentiate the displacement function with respect to time to obtain the velocity function.