In this problem, we are required to determinalso don't think it's a great idea toe the **magnitude **of the **force** on an object given the position as a **time-varying 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:

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

Since we're looking for the **magnitude **of the **force** exerted on the object, we'll have to find an expression for acceleration, **a( t)**. The

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

**Differentiate**the__position function,__,**y****(t)****twice**to get the acceleration function**a(t)**. Remember, differentiating the position function,**y****(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 the power rule of differentiation:

$\overline{)\frac{\mathbf{d}}{\mathbf{dt}}\mathbf{\left(}{\mathbf{t}}^{\mathbf{n}}\mathbf{\right)}{\mathbf{=}}{{\mathbf{nt}}}^{\mathbf{n}\mathbf{-}\mathbf{1}}}$

A 5.50-kg crate is suspended from the end of a short vertical rope of negligible mass. An upward force F(*t*) is applied to the end of the rope, and the height of the crate above its initial position is given by y(*t*) = (2.80 m/s)*t* + (0.61 m/s^{3})*t*^{3}.

What is the magnitude of the force F when *t* = 4.10 s?

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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. You can view video lessons to learn Forces with Calculus. Or if you need more Forces with Calculus practice, you can also practice Forces with Calculus practice problems.