Astronauts in space 'weigh' themselves by oscillating on a spring. Suppose the position of an oscillating 75 kg astronaut is given by x = (0.30 m) sin ((𝝅 rad/s) X t), where t is in s. What force does the spring exert on the astronaut at (a) t = 1.0 s and (b) 1.5 s? Note that the angle of the sine function is in radians.

Verified step by step guidance

1. First, identify the amplitude (A) and angular frequency (\(\omega\)) from the given position function \(x = 0.30 \, \text{m} \cdot \sin(\pi \, \text{rad/s} \cdot t)\). Here, \(A = 0.30 \, \text{m}\) and \(\omega = \pi \, \text{rad/s}\).
2. Calculate the velocity of the astronaut by differentiating the position function with respect to time, \(t\). The velocity function \(v(t)\) is given by \(v(t) = A\omega \cos(\omega t)\).
3. Substitute the values of \(A\) and \(\omega\) into the velocity function to get \(v(t) = 0.30 \, \text{m} \cdot \pi \, \text{rad/s} \cdot \cos(\pi t)\).
4. To find the force exerted by the spring at specific times, use Hooke's Law, which states that the force exerted by a spring is proportional to the displacement from the equilibrium position, \(F = -kx\), where \(k\) is the spring constant. Since the problem does not provide \(k\), focus on the form of the force function, \(F(t) = -kx(t)\).
5. Evaluate the force function at the given times: (a) \(t = 1.0 \, s\) and (b) \(t = 1.5 \, s\) by substituting these values into the position function \(x(t)\) and then using the force function \(F(t)\).

Key Concepts

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. The motion can be described by a sine or cosine function, indicating that the restoring force is proportional to the displacement from the equilibrium. In this case, the astronaut's position is modeled by a sine function, which is characteristic of SHM.

Hooke's Law

Hooke's Law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position, expressed as F = -kx, where k is the spring constant and x is the displacement. This principle is essential for calculating the force exerted by the spring on the astronaut as they oscillate, as it relates the position of the astronaut to the force experienced.

Force Calculation in Oscillatory Motion

In oscillatory motion, the net force acting on an object can be determined by substituting the position function into Hooke's Law. For the given position function of the astronaut, the force exerted by the spring at specific times can be calculated by finding the displacement at those times and applying the spring constant. This allows us to understand how the force varies as the astronaut oscillates.