Q.8
A person sitting inside an elevator performs a weighing experiment with an object of mass 50 kg. The variation of the height $y$ (in m) of the elevator, from the ground, with time $t$ (in s) is given by:

$$\mathrm{<span\; style="background-color:\; white;">y</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">8</span>}<span\; style="background-color:\; white;">[</span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">sin</span>}<span\; style="background-color:\; white;"></span><span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}\mathrm{<span\; style="background-color:\; white;">t</span>}}{\mathrm{<span\; style="background-color:\; white;">T</span>}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">]</span><span\; style="background-color:\; white;">,</span>\text{<span style="background-color: white;">where </span>}\mathrm{<span\; style="background-color:\; white;">T</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">40</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">s</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

Taking the acceleration due to gravity $g=10{\text{m/s}}^2$, the maximum variation of the object’s weight (in N) as observed in the experiment is ______.

Answer: 2

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Solution:

1. Understanding the Problem

• An elevator moves vertically with its height $y$ described by a sinusoidal function:

  $$\mathrm{<span\; style="background-color:\; white;">y</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">8</span>}<span\; style="background-color:\; white;">[</span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">sin</span>}<span\; style="background-color:\; white;"></span><span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}\mathrm{<span\; style="background-color:\; white;">t</span>}}{\mathrm{<span\; style="background-color:\; white;">T</span>}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">]</span><span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">T</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">40</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">s</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

• The object has a mass $m=50\text{kg}$, and $g=10{\text{m/s}}^2$.

• We need to find the maximum variation in the object's apparent weight as observed on a scale inside the elevator.

2. Key Physics Concepts

• Apparent Weight: The normal force $N$ exerted by the scale, which equals the object's apparent weight.

  • When the elevator accelerates upward, $N=m(g+a)$.

  • When the elevator accelerates downward, $N=m(g-a)$.

• The variation in weight is due to the elevator's acceleration $a$, derived from the second time derivative of $y(t)$.

3. Derive the Elevator's Acceleration

Given:

$$\mathrm{<span\; style="background-color:\; white;">y</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">8</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">8</span>}\mathrm{<span\; style="background-color:\; white;">sin</span>}<span\; style="background-color:\; white;"></span><span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}\mathrm{<span\; style="background-color:\; white;">t</span>}}{\mathrm{<span\; style="background-color:\; white;">T</span>}}<span\; style="background-color:\; white;">)</span>\mathrm{<span\; style="background-color:\; white;">.</span>}$$

• First Derivative (Velocity):

  $$\mathrm{<span\; style="background-color:\; white;">v</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">d</span>}\mathrm{<span\; style="background-color:\; white;">y</span>}}{\mathrm{<span\; style="background-color:\; white;">d</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">8</span>}<span\; style="background-color:\; white;">\cdot </span>\frac{\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}}{\mathrm{<span\; style="background-color:\; white;">T</span>}}\mathrm{<span\; style="background-color:\; white;">cos</span>}<span\; style="background-color:\; white;"></span><span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}\mathrm{<span\; style="background-color:\; white;">t</span>}}{\mathrm{<span\; style="background-color:\; white;">T</span>}}<span\; style="background-color:\; white;">)</span>\mathrm{<span\; style="background-color:\; white;">.</span>}$$

• Second Derivative (Acceleration):

  $$\mathrm{<span\; style="background-color:\; white;">a</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">d</span>}\mathrm{<span\; style="background-color:\; white;">v</span>}}{\mathrm{<span\; style="background-color:\; white;">d</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}}<span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">8</span>}{<span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}}{\mathrm{<span\; style="background-color:\; white;">T</span>}}<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\mathrm{<span\; style="background-color:\; white;">sin</span>}<span\; style="background-color:\; white;"></span><span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}\mathrm{<span\; style="background-color:\; white;">t</span>}}{\mathrm{<span\; style="background-color:\; white;">T</span>}}<span\; style="background-color:\; white;">)</span>\mathrm{<span\; style="background-color:\; white;">.</span>}$$

Substitute $T=40\pi$:

$$\mathrm{<span\; style="background-color:\; white;">a</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">8</span>}{<span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}}{\mathrm{<span\; style="background-color:\; white;">40</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}}<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\mathrm{<span\; style="background-color:\; white;">sin</span>}<span\; style="background-color:\; white;"></span><span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}\mathrm{<span\; style="background-color:\; white;">t</span>}}{\mathrm{<span\; style="background-color:\; white;">40</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">8</span>}{<span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">20</span>}}<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\mathrm{<span\; style="background-color:\; white;">sin</span>}<span\; style="background-color:\; white;"></span><span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">t</span>}}{\mathrm{<span\; style="background-color:\; white;">20</span>}}<span\; style="background-color:\; white;">)</span>\mathrm{<span\; style="background-color:\; white;">.</span>}$$$$\mathrm{<span\; style="background-color:\; white;">a</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">8</span>}<span\; style="background-color:\; white;">\cdot </span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">400</span>}}\mathrm{<span\; style="background-color:\; white;">sin</span>}<span\; style="background-color:\; white;"></span><span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">t</span>}}{\mathrm{<span\; style="background-color:\; white;">20</span>}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">-</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">50</span>}}\mathrm{<span\; style="background-color:\; white;">sin</span>}<span\; style="background-color:\; white;"></span><span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">t</span>}}{\mathrm{<span\; style="background-color:\; white;">20</span>}}<span\; style="background-color:\; white;">)</span>\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\text{<span style="background-color: white;">m/s</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

4. Determine Apparent Weight

The apparent weight $W$ varies with the acceleration $a(t)$:

$$\mathrm{<span\; style="background-color:\; white;">W</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">m</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">g</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">a</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">50</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">10</span>}<span\; style="background-color:\; white;">-</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">50</span>}}\mathrm{<span\; style="background-color:\; white;">sin</span>}<span\; style="background-color:\; white;"></span><span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">t</span>}}{\mathrm{<span\; style="background-color:\; white;">20</span>}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">)</span>\mathrm{<span\; style="background-color:\; white;">.</span>}$$

The maximum variation occurs when $\sin \left(\frac{t}{20}\right)$ reaches its extreme values ($\pm 1$):

• Maximum Weight:

  $${\mathrm{<span\; style="background-color:\; white;">W</span>}}_{\text{<span style="background-color: white;">max</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">50</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">10</span>}<span\; style="background-color:\; white;">+</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">50</span>}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">501</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">N</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

• Minimum Weight:

  $${\mathrm{<span\; style="background-color:\; white;">W</span>}}_{\text{<span style="background-color: white;">min</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">50</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">10</span>}<span\; style="background-color:\; white;">-</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">50</span>}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">499</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">N</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

• Variation in Weight:

  $$\mathrm{<span\; style="background-color:\; white;">\Delta </span>}\mathrm{<span\; style="background-color:\; white;">W</span>}<span\; style="background-color:\; white;">=</span>{\mathrm{<span\; style="background-color:\; white;">W</span>}}_{\text{<span style="background-color: white;">max</span>}}<span\; style="background-color:\; white;">-</span>{\mathrm{<span\; style="background-color:\; white;">W</span>}}_{\text{<span style="background-color: white;">min</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">501</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">499</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">2</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">N</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

5. Final Answer

The maximum variation of the object’s weight is:

\boxed{2}