Problem: Astronauts in space "weigh" themselves by oscillating on a spring. Suppose the position of an oscillating 76-kg astronaut is given by x = (0.31 m) sin((π rad/s)t), where t is in s.(a) What force does the spring exert on the astronaut at t = 1.0 s? Note that the angle of the sine function is in radians.(b) What force does the spring exert on the astronaut at t=1.5 s?

FREE Expert Solution

In this problem, we are required to determine the force on an object given the position as a time-varying sine function.

Since we have the position as a function of time, we know this is a calculus problem.

A diagram like this one can help you remember the relationships between the variables:

$\mathbit{P}\begin{array}{c}{\mathbf{←}}\\ {\mathbf{\to }}\end{array}\underset{\frac{\mathbit{d}}{\mathbit{d}\mathbit{t}}}{\overset{{\mathbf{\int }}{\mathbit{d}}{\mathbit{t}}}{\mathbit{V}}}\begin{array}{c}{\mathbf{←}}\\ {\mathbf{\to }}\end{array}\underset{\mathbit{F}\mathbf{=}\mathbit{m}\mathbit{a}}{\mathbit{A}\mathbf{,}\mathbit{F}}$

We're looking for the force exerted on the object. That means we have to find an expression for acceleration, a(t). We'll obtain a(t) by differentiating the position function, x(t), twice then multiply it by the mass, m

The steps needed to solve this problem are simple and straight forward:

1. Differentiate the position function, x(t), twice to get the acceleration function a(t). Remember, differentiating the position function, x(t), once gives the velocity function, v(t).
2. Use the equation ma to find a function for the force
3. Calculate the force at a specific time of interest.

In Step 1, we'll need to remember how to differentiate trigonometric functions.

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Problem Details

Astronauts in space "weigh" themselves by oscillating on a spring. Suppose the position of an oscillating 76-kg astronaut is given by x = (0.31 m) sin((π rad/s)t), where t is in s.
(a) What force does the spring exert on the astronaut at t = 1.0 s? Note that the angle of the sine function is in radians.
(b) What force does the spring exert on the astronaut at t=1.5 s?