Question

A steel ball of mass $m$falls from a height $h$onto a scale calibrated in newtons. The ball rebounds repeatedly to nearly the same height $h$. The scale is sluggish in its response to the intermittent hits and displays an average force$F$avg, such that $F\_\{\backslash text\{avg\left\}\right\}T=FE\_\left\{t\right\},$where $F{y}_t$is the brief impulse that the ball imparts to the scale on every hit, and $\mathrm{T}$is the time between hits.

Question: Calculate this average force in terms of $\mathrm{m},\mathrm{h}$and physical constants. Compare your result with the scale reading if the ball merely rests on the scale. Explain your analysis carefully (but briefly).

Short answer

The value of average force

$F_{\text{evg}}=\frac{2m\sqrt{2gh}}{T}$

Step-by-step solution

Identification of given data

• Mass of steel ball - m

• Height - h

Calculation of the average force.

Striking velocity,$v=\sqrt{2gh}$

Rebound velocity$v=-\sqrt{2gh}$

Then the impulse

$\backslash begin\left\{aligned\right\}\&I=m\backslash timesV\backslash \backslash \&I=m(\backslash sqrt\{2gh\}-(-\backslash sqrt\{2gh\}\left)\right)\backslash \backslash \&I=2m\backslash sqrt\{2gh\}\backslash end\left\{aligned\right\}$

Force,

$\backslash begin\left\{aligned\right\}\&F=\backslash frac\{mV\}\{\backslash text\{ق\}t\}\backslash \backslash \&F=\backslash frac\{2m\backslash sqrt\{2gh\left\}\right\}\{\backslash text\{Rt\left\}\right\}\backslash end\left\{aligned\right\}$

As given in the question

$\backslash begin\left\{aligned\right\}\&F\_\{avg\}T=F\backslash rrbrackett\backslash \backslash \&F\_\{\backslash text\{avg\left\}\right\}T=F冈t\backslash \backslash \&F\_\{\backslash text\{avg\left\}\right\}T=\backslash frac\{2m\backslash sqrt\{2gh\left\}\right\}\{冈t\}\backslash times冈t\backslash \backslash \&F\_\{\backslash text\{avg\left\}\right\}=\backslash frac\{2m\backslash sqrt\{2gh\left\}\right\}\left\{T\right\}\backslash end\left\{aligned\right\}$

Thus, the value of average force becomes

$F_{\text{avg}}=\frac{2m\sqrt{2gh}}{T}$