Question

A body having mass \(5 \mathrm{~g}\) is executing S.H.M. with an amplitude of \(0.3 \mathrm{~m}\). If the periodic time of the system is \((\pi / 10) \mathrm{s}\), then the maximum force acting on body is \(\ldots \ldots \ldots \ldots\) (A) \(0.6 \mathrm{~N}\) (B) \(0.3 \mathrm{~N}\) (C) \(6 \mathrm{~N}\) (D) \(3 \mathrm{~N}\)

Short answer

The maximum force acting on the body is 0.6 N (Option A).

Step-by-step solution

Identify the given values and the formula to find maximum acceleration

We are given the mass (m) of the body, the amplitude (A) of the oscillation, and the periodic time (T). We will use the maximum acceleration (a_max) formula in SHM, which is: a_max = ω²A, where ω is the angular frequency, and it is related to the periodic time by the formula: ω = 2π/T.

Calculate the angular frequency (ω)

Using the formula ω = 2π/T, we can find ω. We are given T = π/10 s, so we have: ω = 2π/(π/10) ω = 2 × 10 ω = 20 rad/s

Calculate the maximum acceleration (a_max)

Now, we can use the maximum acceleration formula: a_max = ω²A. We have found ω and the given amplitude A = 0.3 m, so we can calculate a_max: a_max = (20 rad/s)² × 0.3 m a_max = 400 × 0.3 m a_max = 120 m/s²

Calculate the maximum force (F_max)

Finally, we will use Newton's second law of motion (F = ma) to find the maximum force acting on the body. We have the mass (m = 5 g) and maximum acceleration (a_max = 120 m/s²). First, we need to convert the mass from grams to kilograms: m = 5 g × (1 kg/1000 g) m = 0.005 kg Now, we can calculate the maximum force (F_max): F_max = m × a_max F_max = 0.005 kg × 120 m/s² F_max = 0.6 N So, the maximum force acting on the body is 0.6 N, and the correct option is (A).

Key concepts

Maximum Force Calculation

When a body undergoes simple harmonic motion (SHM), it experiences force due to acceleration. The maximum force on the body is determined using Newton's second law: \( F = ma \), where \( F \) is force, \( m \) is mass, and \( a \) is acceleration.
In this specific context, we use the maximum acceleration of the body in SHM to find the maximum force.

• First, convert the mass of the object from grams to kilograms, as the standard unit of mass in physics calculations is kg.
• The formula for maximum acceleration in SHM is related to angular frequency \( \omega \) and amplitude \( A \): \( a_{max} = \omega^2 A \).

Calculate \( a_{max} \) with given \( \omega \) and amplitude, and then apply it in \( F = ma \) to find the maximum force.
This understanding makes it clear how force dynamics work in harmonic motion systems like springs and pendulums.

Angular Frequency

Angular frequency is a key concept in understanding SHM. It represents the rate of change of the phase of a sinusoidal waveform and is crucial for calculating many elements like velocity and acceleration.

• Angular frequency \( \omega \) is related to the periodic time \( T \), which is the time required for one complete cycle of motion.
• The formula is \( \omega = \frac{2\pi}{T} \), expressing how many radians a system executes per second.

Understanding \( \omega \) helps make sense of how a body oscillates within a system and allows for calculation of parameters like maximum acceleration and velocity.
Finding \( \omega \) is the first step in a host of subsequent calculations necessary for SHM analysis.

Newton's Second Law

Newton's second law is foundational in physics, stating that the force acting on an object is equal to the mass of the object multiplied by its acceleration: \( F = ma \).
In SHM, this relationship helps us determine how external forces affect motion. When maximum acceleration is calculated, as shown with max acceleration \( a_{max} \), we plug it into \( F = ma \) to discover the maximum force.

• This law emphasizes that force and acceleration are directly proportional. Doubling the acceleration with a consistent mass doubles the force.
• By analyzing how mass and acceleration affect force, you can predict motion behavior accurately, a critical element in engineering and physics.

This helps break down complex motion into measurable and predictable models, making problems more approachable and easier to solve.