A mass of 0.5 kg oscillates harmonically with an amplitude.of.0.05 m the force.constant of spring is 10 nm. Calculate its 1 ) period of oscillation 2 ) maximum kinetic energy.of.mass

To solve the problem of a mass oscillating harmonically, we can break it down into two parts: calculating the period of oscillation and determining the maximum kinetic energy of the mass. Let's go through each step methodically.

1. Calculating the Period of Oscillation

The period of oscillation (T) for a mass-spring system can be calculated using the formula:

T = 2π√(m/k)

Where:

• T = period of oscillation
• m = mass (in kg)
• k = spring constant (in N/m)

In this case, we have:

• m = 0.5 kg
• k = 10 N/m

Now, substituting the values into the formula:

T = 2π√(0.5 kg / 10 N/m)

Calculating the value inside the square root:

T = 2π√(0.05)

Now, calculating √(0.05):

√(0.05) ≈ 0.2236

Now, substituting this back into the equation for T:

T ≈ 2π × 0.2236 ≈ 1.404 seconds

2. Finding the Maximum Kinetic Energy

The maximum kinetic energy (KE) of the mass during its oscillation can be calculated using the formula:

KE_max = 0.5 * m * v_max²

To find the maximum velocity (v_max), we can use the relationship between maximum velocity, amplitude (A), and angular frequency (ω):

v_max = A * ω

Where:

• A = amplitude (in m)
• ω = angular frequency (in rad/s)

The angular frequency can be calculated using:

ω = 2π/T

We already calculated T, so:

ω = 2π / 1.404 ≈ 4.47 rad/s

Now, substituting the values for amplitude and angular frequency to find v_max:

v_max = 0.05 m * 4.47 rad/s ≈ 0.2235 m/s

Now we can substitute v_max back into the kinetic energy formula:

KE_max = 0.5 * 0.5 kg * (0.2235 m/s)²

Calculating (0.2235 m/s)²:

(0.2235)² ≈ 0.0503

Now substituting this value into the kinetic energy formula:

KE_max ≈ 0.5 * 0.5 kg * 0.0503 ≈ 0.0126 J

Summary of Results

To summarize:

• The period of oscillation is approximately 1.404 seconds.
• The maximum kinetic energy of the mass is approximately 0.0126 Joules.

This analysis illustrates how the properties of mass and spring constant influence the behavior of oscillating systems, providing a clear understanding of harmonic motion. If you have any further questions or need clarification on any part, feel free to ask!