Problem 575 - JEE Physics

Question

Two point masses of \(0.3 \mathrm{~kg}\) and \(0.7 \mathrm{~kg}\) are fixed at the ends of a rod of length \(1.4 \mathrm{~m}\) and of negligible mass. The rod is set rotating about an axis perpendicular to its length with a uniform angular speed. The point on the rod through which the axis should pass in order that the work required for rotation of the rod is minimum, is located at a distance of ..... \(\\{\mathrm{A}\\} 0.4 \mathrm{~m}\) from mass of \(0.3 \mathrm{~kg}\) \\{B \(0.98 \mathrm{~m}\) from mass of \(0.3 \mathrm{~kg}\) \\{C\\} \(0.7 \mathrm{~m}\) from mass of \(0.7 \mathrm{~kg}\) \\{D \(\\} 0.98 \mathrm{~m}\) from mass of \(0.7 \mathrm{~kg}\)

Step-by-Step Solution

Answer

The correct answer is \(\\{\mathrm{B}\\} 0.98 \mathrm{~m}\) from mass of \(0.3 \mathrm{~kg}\).

Step 1: Determine the moment of inertia

Moment of inertia, denoted by I, is the property of an object to resist change in its angular motion. For a point mass rotating about an axis parallel to its length, the moment of inertia is given by the mass times the square of the distance from the axis to the point mass, i.e. \(I = md^2\). In our case, we have two point masses, so the net moment of inertia would be the sum of the individual moments of inertia. For mass \(m_1 = 0.3 \mathrm{~kg}\), let the distance from the axis to the mass be d. The moment of inertia for this mass is \(I_1 = m_1 d^2\). For mass \(m_2 = 0.7 \mathrm{~kg}\), since the total length of the rod is \(1.4 \mathrm{~m}\), the distance from the axis to this mass would be \(1.4 \mathrm{~-} d\). The moment of inertia for this mass is \(I_2 = m_2 (1.4 - d)^2\). The net moment of inertia is given by \(I = I_1 + I_2\).

Step 2: Differentiate the moment of inertia with respect to distance

Next, we need to minimize the moment of inertia I with respect to the distance d. To do this, we'll differentiate I with respect to d and set the derivative equal to 0. Therefore, \(\frac{dI}{dd} = \frac{d}{dd}(0.3d^2 + 0.7(1.4-d)^2) = 0\).

Step 3: Solve for distance

Now, we need to solve the equation obtained in step 2 for the distance d. \(\frac{d}{dd}(0.3d^2 + 0.7(1.4-d)^2) = 0.6d - 1.4(0.7) + 1.4(0.7)d = 0\) Simplifying and solving for d, we get: \(d(d(0.6 + 0.7\cdot1.4) = 1.4(0.7)\) \(d = \frac{1.4(0.7)}{0.6 + 0.7(1.4)}\) \(d = 0.98 \mathrm{~m}\) According to the answer, the distance of 0.98 m is from mass \(0.3 \mathrm{~kg}\), which corresponds to option B. Thus, the correct answer is: \(\\{\mathrm{B}\\} 0.98 \mathrm{~m}\) from mass of \(0.3 \mathrm{~kg}\)

Key Concepts

Point Masses

A point mass is a simplified model where all the mass of an object is assumed to be concentrated at a single point. Point masses are useful in physics, particularly when analyzing objects with rotational motion. In our exercise, we consider two point masses attached to a rod: one with a mass of \(0.3 \, \mathrm{kg}\) and the other \(0.7 \, \mathrm{kg}\). These masses are placed at each end of the rod, which is considered to have negligible mass. This allows us to focus our calculations on the mass distribution along the rod.
Understanding point masses helps simplify the computation of the moment of inertia—which is crucial in determining how the rod will behave when it rotates about an axis. By concentrating mass at discrete points, we can easily apply physical principles that rely on distance from the axis and mass. This also simplifies solving for rotational properties such as angular acceleration and the amount of work needed to rotate the system.

Rotational Motion

Rotational motion occurs when an object spins around an axis. In the case of the exercise, the rod rotates around an axis perpendicular to its length. This kind of motion is crucial in understanding how different forces and factors such as moment of inertia affect how objects move.

• When discussing rotational motion, the placement of the axis plays a critical role. The distance of point masses from the axis directly influences the moment of inertia, which measures resistance to changes in rotational speed.
• Just like linear motion, rotational motion explains how objects gain speed, slow down, and maintain their state due to forces applied to them.

Understanding rotational motion and connecting it to point masses is essential for solving physics problems involving such systems.

Angular Speed

Angular speed is the rate at which an object rotates or revolves relative to another point, typically the center of a rotating system. It is measured in radians per second in most systems.
In this exercise, the rod rotates with a uniform angular speed, meaning it spins at a constant rate. This uniformity provides a stable basis for calculating physical properties like moment of inertia and assessing how different mass placements affect the system. Uniform angular speed means that every point on the rod has the same rate of rotation around the axis of rotation.
Key points about angular speed:

• Angular speed is directly related to how far each mass is from the rotation axis; changing the distance affects the rotational dynamics.
• Understanding angular speed is crucial for designing systems that require balance, such as machinery and vehicles.

Overall, knowing how angular speed operates in a system helps in designing and optimizing rotational devices.

Minimizing Work

Minimizing the work required to rotate an object is an essential consideration in mechanical design. In this exercise, the goal is to find the point where an axis should pass through the rod to minimize the work done in turning the rod.

• Work in rotational dynamics is related to the moment of inertia; a lower moment of inertia means less force is needed to achieve the same rotational speed.
• By adjusting the axis position, we can reduce unnecessary effort and energy consumption when maintaining or changing the angular speed.

In our exercise, solving for the optimal distance along the rod helps ensure that the setup is as efficient as possible. By finding the specific location, in this case, \(0.98 \, \mathrm{m}\) from the \(0.3 \, \mathrm{kg}\) mass, the workload required to reach the desired angular speed is minimized, showcasing the balance between mass distribution and mechanical efficiency.