Question 11: Properties of Matter

Question

A lift of mass 1600 kg is supported by thick iron wire. If the maximum stress which the wire can withstand is $4 	imes 10^8 \mathrm{~N} / \mathrm{m}^2$ and its radius is 4 mm , then maximum acceleration the lift can take is $\_\_\_\_$ $\mathrm{m} / \mathrm{s}^2$.
(take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2$ and $\pi=3.14$ )

Accepted Answer

When a lift of mass M accelerates upward with acceleration a, the tension (T) in the supporting cable must overcome both gravity and provide the upward acceleration.
Using Newton's Second Law ( $\mathrm{F}_{	ext {net }}=\mathrm{Ma}$ ):
$$ \mathrm{T}-\mathrm{Mg}=\mathrm{Ma} \Rightarrow \mathrm{~T}=\mathrm{M}(\mathrm{~g}+\mathrm{a}) $$
The stress $(\sigma)$ is defined as the restoring force (tension) per unit cross-sectional area (A):
$$ \sigma=\frac{T}{A}=\frac{M(g+a)}{A} $$
For a wire with a circular cross-section of radius r , the area of cross-section is $\mathrm{A}=\pi \mathrm{r}^2$.
$$ \sigma_{\max }=\frac{\mathrm{M}\left(\mathrm{~g}+\mathrm{a}_{\max }ight)}{\pi \mathrm{r}^2} $$
The mass of lift is $\mathrm{M}=1600 \mathrm{~kg}$
The radius of cable is $\mathrm{r}=4 \mathrm{~mm}=4 	imes 10^{-3} \mathrm{~m}$
So, the area is
$$ \mathrm{A}=\pi \mathrm{r}^2=3.14 	imes\left(4 	imes 10^{-3}ight)^2=3.14 	imes 16 	imes 10^{-6} \mathrm{~m}^2 $$
$$\Rightarrow $$ $$ \mathrm{A}=50.24 	imes 10^{-6} \mathrm{~m}^2 $$
So, the maximum stress is
$$ \sigma_{\max }=4 	imes 10^8 \mathrm{~N} / \mathrm{m}^2 $$
From the maximum stress formula,
$$ 4 	imes 10^8=\frac{1600\left(10+\mathrm{a}_{\max }ight)}{50.24 	imes 10^{-6}} $$
$$\Rightarrow $$ $4 	imes 10^8 	imes 50.24 	imes 10^{-6}=1600\left(10+a_{\max }ight)$
$$\Rightarrow $$ $200.96 	imes 10^2=1600\left(10+a_{\max }ight)$
$$\Rightarrow $$ $20096=1600\left(10+a_{\max }ight)$
$$\Rightarrow $$ $10+a_{\max }=\frac{20096}{1600}=12.56$
$$\Rightarrow $$ $$ \mathrm{a}_{\max }=12.56-10=2.56 \mathrm{~m} / \mathrm{s}^2 $$
Therefore, the maximum acceleration of the lift is $2.56 \mathrm{~m} / \mathrm{s}^2$.
Thus, the correct option is (A).

Answer

2.56

Answer

3.89

Answer

4.32

Answer

5.16

Properties of Matter

Choose the correct answer: