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:

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

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

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

**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)**.**Use the equation***F*=to find a function for the**ma**__force__**Calculate the force**at a specific time of interest.

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

$\overline{)\frac{\mathbf{d}}{\mathbf{dt}}{\mathbf{sin}}{\mathbf{}}{\mathbf{\left(}}{\mathbf{\theta}}{\mathbf{t}}{\mathbf{\right)}}{\mathbf{=}}{\mathbf{\theta}}{\mathbf{}}{\mathbf{cos}}{\mathbf{}}{\mathbf{\left(}}{\mathbf{\theta}}{\mathbf{t}}{\mathbf{\right)}}\phantom{\rule{0ex}{0ex}}\frac{\mathbf{d}}{\mathbf{dt}}{\mathbf{}}{\mathbf{cos}}{\mathbf{}}{\mathbf{\left(}}{\mathbf{\theta}}{\mathbf{t}}{\mathbf{\right)}}{\mathbf{=}}{\mathbf{-}}{\mathbf{\theta}}{\mathbf{}}{\mathbf{sin}}{\mathbf{}}{\mathbf{\left(}}{\mathbf{\theta}}{\mathbf{t}}{\mathbf{\right)}}}$

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?

Frequently Asked Questions

What scientific concept do you need to know in order to solve this problem?

Our tutors have indicated that to solve this problem you will need to apply the Forces with Calculus concept. If you need more Forces with Calculus practice, you can also practice Forces with Calculus practice problems.