Question

One end of a mass less spring having force constant \(\mathrm{k}\) and length \(50 \mathrm{~cm}\) is attached at the upper end of a plane inclined at an angle \(e=30^{\circ} .\) When a body of mass \(m=1.5 \mathrm{~kg}\) is attached at the lower end of the spring, the length of the spring increases by \(2.5 \mathrm{~cm}\). Now, if the mass is displaced by a small amount and released, the amplitude of the resultant oscillation is ......... (A) \((\pi / 7)\) (B) \((2 \pi / 7)\) (C) \((\pi / 5)\) (D) \((2 \pi / 5)\)

Short answer

The amplitude of the resultant oscillation is approximately 0.0125 m, which does not match any of the given answer choices.

Step-by-step solution

Identify the forces acting on the mass

When the mass is attached to the spring, it feels the spring force (Fs), the force due to gravity (Fg), and the normal force (Fn) from the inclined plane. As we are focusing on the oscillation of the mass in the direction of the spring's elongation on the inclined plane, the relevant forces are the spring force and the component of the gravitational force acting in that direction.

Establish the spring force equation

The spring force (Fs) follows Hooke's law, which states that the force exerted by the spring is proportional to its elongation (Δx) and the spring constant (k): Fs = k(Δx) In this case, θ = 30°, the spring constant k is given, the elongation Δx is 0.025 m, and the mass m = 1.5 kg.

Find the component of gravitational force acting along the spring

To find this component, we use the angle θ (30°) and the gravitational force acting on the mass (Fg = mg, g being the acceleration due to gravity): Fg_comp = Fg * sin(θ) Fg_comp = (1.5 kg)(9.81 m/s²) * sin(30°) Fg_comp ≈ 7.35 N

Calculate the spring constant using the spring force equation

Given that the spring is in equilibrium when the mass is initially attached, the spring force equals the component of the gravitational force acting along the spring: Fs = Fg_comp k(Δx) = Fg_comp k = Fg_comp/Δx k ≈ 7.35 N / 0.025 m k ≈ 294 N/m

Calculate the amplitude of the oscillation

The amplitude A of the oscillation can be found using the following equation for a simple harmonic motion: A = Δx * sin(θ) In our case, Δx = 0.025 m and θ = 30°, so we have: A = (0.025 m) * sin(30°) A ≈ 0.0125 m Comparing the calculated amplitude with the given options: (A) \(\frac{\pi}{7}\) ≈ 0.4488 (B) \(\frac{2\pi}{7}\) ≈ 0.8978 (C) \(\frac{\pi}{5}\) ≈ 0.6283 (D) \(\frac{2\pi}{5}\) ≈ 1.2566 None of the given options match the calculated amplitude exactly. Since the problem statement asks for the amplitude of the "resultant oscillation," it is possible that the correct answer is not provided, or there may be a rounding error.

Key concepts

Hooke's Law

Hooke's Law is a fundamental principle that describes how elastic materials like springs behave under force. The law states that the force exerted by a spring is proportional to the displacement or elongation it experiences. In equation form, this is expressed as:\[F_s = -k \cdot \Delta x\]where:

• \(F_s\) is the spring force.
• \(k\) is the spring constant, a measure of the spring's stiffness.
• \(\Delta x\) is the change in length of the spring from its equilibrium position.

The negative sign indicates that the force exerted by the spring is in the opposite direction of the displacement. The spring constant \(k\) is specific to each spring and determines how much force is needed to extend or compress the spring by a certain distance.
In our exercise, we calculated \(k\) to be approximately \(294 \text{ N/m}\), using the given elongation and gravitational force component.

Gravitational Force Component

The gravitational force component along an inclined plane is a crucial factor in analyzing motion involving such planes. Since gravity acts vertically downwards, only a portion of this force helps move an object up or down the incline.
To determine this component, consider the angle \(\theta\) of the incline. Use trigonometry to find the component of the gravitational force parallel to the slope:\[F_{g\_comp} = mg \cdot \sin\theta\]where:

• \(F_{g\_comp}\) is the component of the gravitational force.
• \(mg\) is the weight of the object, with \(g\) being approximately \(9.81 \text{ m/s}^2\).
• \(\theta\) is the angle of the incline.

For our problem, we found that the component of the gravitational force acting along the plane was approximately \(7.35 \text{ N}\) when \(\theta = 30^\circ\). This component is instrumental since it counterbalances the spring force when the mass is in equilibrium on the inclined plane.

Inclined Plane Dynamics

Inclined plane dynamics focuses on objects moving on sloped surfaces, which is common in physics problems. To analyze such systems, it's essential to consider the forces acting along the slope and across it.
Benefits of studying inclined planes:

• Understand the balance of forces including gravity, normal force, and friction where applicable.
• Learn how to calculate components of forces, such as gravitational force, which affect the object's motion.

For this system, the main forces include:

• Gravity, providing both a perpendicular and parallel component to the plane.
• The spring force, acting opposite to the direction of elongation of the spring along the plane.
• Normal force, acting perpendicular to the inclined surface, counteracting gravity's perpendicular component.

When an object on an inclined plane is attached to a spring, like in our scenario, it can oscillate back and forth if displaced. This kind of motion can be resolved using principles from simple harmonic motion, where understanding how forces dictate motion helps in predicting behaviors such as amplitude and frequency of oscillation.
This is crucial for solving problems akin to determining the resultant oscillation amplitude like in our initial task.