Problem 467 - JEE Physics

Question

A particle of mass \(0.1 \mathrm{~kg}\) is subjected to a force which varies with distance as shown in figure. If it starts its journey from rest at \(\mathrm{x}=0\). What is the particle's velocity square at \(\mathrm{x}=6 \mathrm{~cm}\) ? (A) \(0(\mathrm{~m} / \mathrm{s})^{2}\) (B) \(240 \sqrt{2}(\mathrm{~m} / \mathrm{s})^{2}\) (C) \(240 \sqrt{3}(\mathrm{~m} / \mathrm{s})^{2}\) (D) \(480(\mathrm{~m} / \mathrm{s})^{2}\)

Step-by-Step Solution

Answer

The square of the particle's velocity at x=6 cm is \(-24 (\mathrm{~m} / \mathrm{s})^{2}\). This negative value indicates an inconsistency in the given values or the figure, and doesn't match any of the given options (A, B, C, or D).

Step 1: Convert the given distance to SI units

Convert the given distance x = 6 cm into meters (SI unit for distance) by dividing it by 100: \[x = \frac{6}{100} \mathrm{m} = 0.06 \mathrm{m}\]

Step 2: Calculate the area under the force-displacement curve

We can find the area under the force-displacement curve by dividing the entire area into triangles and rectangles. The figure shows a right triangle with its base along the x-axis and its height along the y-axis. The base has a length of 0.06 m (from step 1) and the height is given by the force at x=0.06 m, that is, \(F = -40 \mathrm{~N}\). The negative sign indicates that the force opposes the displacement. The area of a triangle is given by \(\frac{1}{2} \times base \times height\), thus the area under the curve (which represents work done) is \[W = -\frac{1}{2} \times 0.06 \mathrm{m} \times 40\mathrm{~N} = -1.2\mathrm{~J}\]

Step 3: Use the work-energy theorem to calculate the change in kinetic energy

According to the work-energy theorem, the work done on the particle equals the change in its kinetic energy. The particle starts its journey from rest, so the initial kinetic energy is \(0 \mathrm{~J}\). Let the final kinetic energy be \(KE_f\), then \[W = KE_f - KE_i\] Since the initial kinetic energy is 0, we have \[W = KE_f\] Substitute the work done value found in step 2: \[-1.2\mathrm{~J} = KE_f\]

Step 4: Calculate the particle's velocity square using the Kinetic energy expression

The kinetic energy of the particle is given by \(\frac{1}{2}mv^2\), where m is the particle's mass and v is its velocity. Now we have \(KE_f = -1.2 \mathrm{~J}\), and the particle's mass is \(0.1 \mathrm{~kg}\). Solving for the particle's velocity square, we get \[\frac{1}{2}(0.1 \mathrm{~kg})v^2 = -1.2 \mathrm{~J}\] \[v^2 = \frac{-2 \times 1.2 \mathrm{~J}}{0.1 \mathrm{~kg}}\] \[v^2 = -24 (\mathrm{~m} / \mathrm{s})^{2}\] The square of the particle's velocity at x=6 cm is \(-24 (\mathrm{~m} / \mathrm{s})^{2}\). Since this value is negative, it indicates that there's an inconsistency in the given values or the figure. It could be a mistake in the problem statement or a misinterpretation of the figure. This result doesn't match any of the given options (A, B, C, or D).

Key Concepts

Force-Displacement Curve

The force-displacement curve is a graphical representation where the force applied to an object is plotted against the displacement that this force causes.
This kind of curve is essential in analyzing the work done by a force over a distance. To find the work done on an object from the force-displacement curve, you must calculate the area under the curve. This area represents the total energy transferred to the object by the force. It can include different shapes like triangles and rectangles, each requiring its formula to compute the area.

• The area under the curve gives positive work when the force helps in the direction of the displacement.
• Negative work occurs when the force opposes the displacement.

By comprehending this concept, you can determine the mechanical work and thereby predict the change in the object's energy, which links directly to the work-energy theorem.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. This type of energy is directly proportional to both the mass of the object and the square of its velocity. The fundamental equation is given by: \[ KE = \frac{1}{2}mv^2 \]Where:

• \( KE \) is the kinetic energy,
• \( m \) is the mass,
• \( v \) is the velocity of the object.

Even a tiny object can have a significant amount of kinetic energy if it moves fast enough. In the context of the work-energy theorem, the work done on an object results in a change in its kinetic energy. This principle allows us to calculate the object's velocity if certain conditions are known, such as its initial state and the work involved.

Particle Dynamics

Particle dynamics involves the study of forces and the resultant motion they cause on particles. It forms the core of classical mechanics, focusing on how an object's movement can change over time. In dynamics, understanding how a force affects a particle's motion is crucial. This includes recognizing how different forces interact and how they might balance out or disturb the system. Key aspects include:

• Understanding net forces and how they translate into acceleration via Newton's second law \( F = ma \).
• Recognizing the impact of resistive forces, which might include friction or as seen in the exercise, forces opposing the motion.
• Applying concepts like momentum and energy conservation to predict the outcomes of various interactions.

By analyzing particle dynamics, we gain insights into not only how things move but why they move the way they do. In solving problems, having a solid grasp on these principles is essential for extracting accurate solutions.