Question

Consider the block–spring–surface system in part (B) of Example 8.6. (a) Using an energy approach, find the position x of the block at which its speed is a maximum. (b) In the What If? Section of this example, we explored the effects of an increased friction force of 10.0 N. At what position of the block does its maximum speed occur in this situation?

Short answer

(a) The position of the block, when friction between block and surface is 4N , is $x=-4.0\times {10}^{-3}m$

(b) The position of the block, when friction between block and surface is 10N , is.

$x=-{10}^{-2}m.$

Step-by-step solution

Introduction

In the presence of non-conservative forces, the law of conservation of energy states that the sum of the mechanical energy of a body and the total work done by the dissipative forces, remains constant.

$\mathbf{∆}\mathit{K}\mathbf{+}\mathbf{∆}\mathit{U}\mathbf{+}\mathbf{∆}{\mathit{E}}_{\mathbf{i}\mathbf{n}\mathbf{t}}\mathbf{=}\mathbf{0}$

Where,

$\mathbf{∆}\mathit{U}\mathbf{=}$Change in Potential energy

$\mathbf{∆}\mathit{K}\mathbf{=}$Change in kinetic energy

$\mathbf{∆}{\mathit{E}}_{\mathbf{i}\mathbf{n}\mathbf{t}}\mathbf{=}$ Change in internal energy

Calculation for part (a)

As, non-conservative force is acting. Therefore, Equation 8.2 becomes:

$\begin{array}{c}\Delta K+\Delta U+\Delta E_{\mathrm{int}}=0\\ \left(\frac{1}{2}m{v}^2-0\right)+\left(\frac{1}{2}k{x}^2-\frac{1}{2}k{x}_i^2\right)+f_k\left(x_i-x\right)=0\end{array}$

To find the maximum speed, differentiate the equation with respect to x:

$m\frac{dv}{dx}+kx-f_k=0$

Now set $\frac{dV}{dx}=0$

$\begin{array}{c}kx-f_k=0\\ x=\frac{f_k}{k}=\frac{-4.0\text{N}}{1.0\times {10}^3\text{N/m}}=-4.0\times {10}^{-3}\text{m}\end{array}$

This is the compression in the spring, so the position of the block relative to x=0 is

$x=-4.0\times {10}^{-3}\text{m}$

Calculation for part (a)

By the same approach,

$\begin{array}{c}kx-f_k=0\\ x=\frac{f_k}{k}\\ x=\frac{-10.0\text{N}}{1.0\times {10}^3\text{N/m}}\\ x=-{10}^{-2}\text{m}\end{array}$

So the position of the block is

$x=-{10}^{-2}\text{m}$