Astronauts in space cannot weigh themselves by standing on a...

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 P14.36. a. What is her mass if the spring constant is $240 \mathrm{N} / \mathrm{m} ?$ b. What is her speed when the spring's length is $1.2 \mathrm{m} ?$ (FIGURE CANT COPY)

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 P14.36.
a. What is her mass if the spring constant is $240 \mathrm{N} / \mathrm{m} ?$
b. What is her speed when the spring's length is $1.2 \mathrm{m} ?$
(FIGURE CANT COPY)

Key Concepts

Simple Harmonic Motion (SHM)

SHM describes oscillatory motion where the restoring force is directly proportional to the displacement from an equilibrium position and is directed towards that equilibrium. This motion is characterized by sinusoidal oscillations and a constant period that depends only on the system's mass and the spring constant, not on the amplitude of the oscillation.

Spring Constant

The spring constant, denoted by k, is a measure of a spring's stiffness. It relates the force exerted by the spring to its displacement from equilibrium according to Hooke’s Law. In oscillatory systems, the spring constant directly influences the angular frequency and period of the oscillation.

Mass Determination Using Oscillations

In a system undergoing SHM, the mass of the oscillating object can be determined by measuring the period of the oscillation and using the relationship T = 2??(m/k). Since the period is independent of the amplitude in ideal conditions, this method provides a reliable way to determine mass without direct weight measurements.

Conservation of Energy in Oscillatory Systems

This concept involves the principle that the total mechanical energy in a frictionless oscillatory system is conserved, transitioning between potential energy stored in the spring and kinetic energy of the moving mass. By equating these energy forms at different points in the oscillation, one can determine variables such as speed, displacement, or mass.

Kinematics of Oscillatory Motion

This involves understanding the relationships between displacement, velocity, and acceleration in oscillatory motion. In SHM, the maximum speed occurs at the equilibrium position and is related to the amplitude and angular frequency of the oscillation. The precise form of these relationships allows the calculation of instantaneous speed at a given displacement using energy conservation or kinematic equations.

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