Problem 602 - JEE Physics

Question

A meter stick of mass \(400 \mathrm{gm}\) is pivoted at one end and displaced through an angle 600 the increase in its P.E. is \(\overline{\\{\mathrm{A}\\} 2}\) \(\\{B\\} 3\) \(\\{\) C \(\\}\) Zero \(\\{\mathrm{D}\\} 1\)

Step-by-Step Solution

Answer

The increase in potential energy of the meter stick is approximately 1.703 J, which is closest to \(\overline{\mathrm{A}} 2\).

Step 1: Convert mass to kg and angle to radians

Firstly, convert the mass of the meter stick from grams to kilograms: \( mass = \dfrac{400~\mathrm{grams}}{1000} = 0.4 ~\mathrm{kg} \) Now convert the angle of displacement from 60 degrees to radians: \( \theta = 60^\circ * \frac{\pi}{180^\circ} = \frac{\pi}{3} ~\mathrm{radians} \)

Step 2: Calculate the change in height

We need to find out the change in height of the meter stick's center of mass due to pivoting. When it is horizontal, the center of mass is right at the middle of the stick (0.5 meters). When it pivots, it moves upwards to form a 30-60-90 degree triangle. The height of the triangle is given by: \( h = 0.5 \times \sin{(\pi/3)} \)

Step 3: Calculate the increase in potential energy

To find the increase in potential energy (ΔP.E.), we need to multiply the mass, gravitational acceleration (g = 9.81 m/s^2), and the change in height: ΔP.E. = mass × g × h ΔP.E. = 0.4 × 9.81 × (0.5 × \(\sin{(π/3)}\)) ΔP.E. ≈ 1.703 J (approx)

Step 4: Choose the correct option

Now that we have calculated the increase in potential energy (ΔP.E.), we can choose the correct option among the given choices. The answer is closest to 2, so the correct option is: \(\overline{\mathrm{A}} 2\)

Key Concepts

Pivoted Meter Stick

A pivoted meter stick is essentially a familiar tool for scientists to study rotational movement and balance. When a meter stick is pivoted, it means one end is fixed, allowing it to rotate around that point. This fixture creates an axis for rotational motion, dividing the possibilities into different angles of inclination. The meter stick has its center of mass located at its midpoint when it is horizontal. This aspect is crucial because, as the stick rotates, the position of the center of mass changes, affecting how potential energy is calculated.
When dealing with a pivoted meter stick, it is important to consider:

• The gravitational forces acting on the stick around the pivot point.
• How the rotation alters the position of the center of mass.

Understanding a pivoted meter stick involves seeing it not just as a simple object, but as a dynamic example of principles of physics in motion.

Angle Conversion

Angle conversion is a key step when working with problems involving rotational motion. Angles can be expressed in different units: degrees and radians. It's essential to understand how to change degrees to radians, as many calculations in physics are made using radians.
The conversion formula for degrees to radians is:

• Multiply the angle in degrees by \( \frac{\pi}{180}\) to get the angle in radians.

For example, to convert 60 degrees to radians:

• 60 \(^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{\pi}{3} \) radians

This concept is useful because it frames the scope and context of problems in a way that integrates well with formulas for potential energy and other physical calculations.

Change in Height

When a pivoted object rotates, like the meter stick, the center of mass changes its vertical position due to the rotation. Calculating this change in height is crucial for finding potential energy changes.
The change in height of an object, in this case during a rotation, often involves geometry. Here, as the meter stick pivots, it forms a triangle with the pivot point and the moved position of the center of mass. The trigonometric relation for height \(h\) in a right triangle is given by:

• \(h = 0.5 \times \sin{(\frac{\pi}{3})}\)

Where 0.5 meters is the horizontal distance of the center of mass from the pivot. This height gives the vertical change necessary to evaluate changes in potential energy.

Potential Energy Calculation

Calculating the increase in potential energy involves understanding that as an object is lifted, work is done against gravitational force. This change is summarized by the formula:

• \(\Delta \text{P.E.} = \text{mass} \times \text{g} \times \text{change in height}\)

For the meter stick:

• Mass = 0.4 kg
• g (gravitational acceleration) = 9.81 \(\text{m/s}^2\)
• Change in height from the previous calculation \(h\)

Substitute into the formula:

• \(\Delta \text{P.E.} = 0.4 \times 9.81 \times (0.5 \times \sin{(\frac{\pi}{3})})\)

The result, approximately 1.703 Joules, indicates how much potential energy increases as the meter stick elevates its mass during the pivot.