Question

If a chain is lowered at a constant speed of $v=1.2\text{m/s}$, determine the normal reaction exerted on the floor as a function of time. The chain has a mass of $80\text{kg}$ and a total length of $6\text{m}$. [Take $g=10{\text{m/s}}^2$]

• A. $19.2\text{N}$
• B. $(160t+19.2)\text{N}$
• C. $(160t+16)\text{N}$
• D. $160t\text{N}$

Answer 1

The correct option is B $(160t+19.2)\text{N}$

Linear mass density of chain $\lambda =\frac{80}{6}=\frac{40}{3}\text{kg/m}$
After time ${}^{\prime }{t}^{\prime }$, length of the portion of the chain lying on the ground $x=1.2t$


Therefore, weight of the portion on the ground
$w=mg=\lambda x\times g=(\frac{40}{3}\times 1.2t)\times 10$
$=160t\text{N}$

Thrust force $F_t=v\times \frac{dm}{dt}$
$[\text{Here}\frac{dm}{dt}=\lambda \cdot \frac{dx}{dt}=\lambda v]$
$\Rightarrow F_t=v\cdot \left(\lambda v\right)$
$=1.2\times \frac{40}{3}\times 1.2$
$=19.2\text{N}$
Normal reaction exerted on floor $=w+F_t$
$=(160t+19.2)\text{N}$ vertically downward.