A metal wire of length 0.5 m and cross-sectional area 10–4 m2 has breaking stress 5 × 108 Nm–2. A block of 10 kg is attached at one end of the string and is rotating in a horizontal circle. The maximum linear velocity of block will be _____ ms–1
Correct Answer: 50
Approach Solution - 1
The problem involves determining the maximum linear velocity of a block tied to a wire before the wire breaks. First, calculate the force that can be sustained by the wire before breaking, using the formula for breaking stress:
\(Breaking\ Stress = \frac{Breaking\ Force}{Cross-sectional\ Area}\)
Given breaking stress, \(5 × 10^8\ N/m^2\), and cross-sectional area, \(10^{-4}\ m^2\), calculate the breaking force:
\(Breaking\ Force = Breaking\ Stress × Cross-sectional\ Area = 5 × 10^8\ N/m^2 × 10^{-4}\ m^2 = 5 × 10^4\ N\)
This force is the maximum tension the wire can withstand. In the case of circular motion, centripetal force is provided by this tension:
\(Centripetal\ Force = \frac{mv^2}{r}\)
where m is the mass (10 kg) and r is the radius (length of the wire = 0.5 m). Set the maximum tension equal to the centripetal force to find the maximum velocity v:
\(5 × 10^4\ N = \frac{10 \times v^2}{0.5}\)
Simplify the equation to solve for v:
\(v^2 = \frac{5 × 10^4 \times 0.5}{10} = 2500\)
Thus, v is calculated as follows:
\(v = \sqrt{2500} = 50\ m/s\)
The computed maximum linear velocity of the block is 50 m/s, which fits within the range [50, 50].
Approach Solution -2
A=10–4m2
l=\(\frac{1}{2} \)m
σ=5×108
\(\frac{mv^2}{la}=5×10^8\)
\(v=\sqrt{\frac{5×10^8×\frac{1}{2}×10^{−4}}{10}}\)
v=5×10=50 m/s
So, the answer is 50 m/s.
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Concepts Used:
Occurrence of metals
The existence of metals often plays an important role in our survival, even if we are unaware of them. Our reliance on metals has been strong since the old industrial era. From buildings to jewelry, we utilize them for everything. But where do we acquire these metals from and how are they made? Let's read the occurrence of metal in more detail.
Metals are found in the earth's crust, and it seems to be a good place to extract them through various processes such as metallurgy. The process of extracting metals from ores is called metallurgy. Mining is the most common method of extracting metals from the earth.
In nature, metals are mostly found in mixed conditions. However, they can also be found in a free state. In its natural state, a native metal is one that can be found in its pure form or as part of an alloy. Most metals are exposed to natural processes such as oxidation and corrosion. As a result, only non-reactive metals such as gold, silver, platinum, and others are discovered in their native or open-free condition. Most metals are obtained as compounds, which must be filtered to remove contaminants before being employed in numerous applications.