Grade 11Mechanics

A partlcle of mass 40 gms executes a s\mp\e harmon\c\ motion of amplitude 2.0 cm . 1f the t\me per\od \s 0.20 seconds, the total mechanlcal energy of the system \s (1)79 X 10‘33 , _ . (2) 7.9 x 10-31 (3) 7-93 . . ~ (4) 793 `

Nitin Kumar

9 Years agoGrade 11

1 Answer

Askiitians Tutor Team

Approved Tutor Answer1 Year ago

To find the total mechanical energy of a particle executing simple harmonic motion (SHM), we can use the formula for total mechanical energy, which is given by:

Total Mechanical Energy in SHM

The total mechanical energy (E) in simple harmonic motion is calculated using the formula:

E = (1/2) k A²

Where:

E = total mechanical energy
k = spring constant
A = amplitude of motion

However, we need to find the spring constant (k) first. The spring constant can be related to the mass (m) and the angular frequency (ω) of the motion. The angular frequency is given by:

ω = 2π/T

Where:

T = time period

Step-by-Step Calculation

Let's break down the steps to find the total mechanical energy:

Convert the mass from grams to kilograms:

m = 40 g = 0.040 kg

Calculate the angular frequency (ω):

Given T = 0.20 seconds, we can find ω:

ω = 2π/T = 2π/0.20 = 10π rad/s

Now, we can find the spring constant (k) using the formula:

k = mω²

Substituting the values:

k = 0.040 kg * (10π)²

k = 0.040 * 100π² ≈ 0.040 * 986.96 ≈ 39.4784 N/m

Now, we can calculate the total mechanical energy:

Using the amplitude A = 2.0 cm = 0.02 m:

E = (1/2) k A² = (1/2) * 39.4784 * (0.02)²

E = (1/2) * 39.4784 * 0.0004 = 0.00789568 J

Final Result

To express this energy in scientific notation:

E ≈ 7.89568 x 10^-3 J

Rounding this to two significant figures gives us:

E ≈ 7.9 x 10^-3 J

Looking at the options provided, the closest match is:

(2) 7.9 x 10^-31 J
(3) 7.93 J

However, it seems there might be a misunderstanding in the options, as our calculated energy is significantly larger than the options provided. The correct answer based on our calculations is approximately 7.9 x 10^-3 J, which does not match any of the given choices. Please double-check the options or the values provided in the question.