The kinetic energy of a body executing S.H.M. is 1/3 of the potential energy. Then, the displacement of the body is x percent of the amplitude, where x isa)33b)50c)67d)87Correct answer is option 'A'. Can you explain this answer?

Question

Answer

We know that PE + KE = TE = constant
Hence at the extremis TE = PE = ½ kz²
Where z is amplitude and k is shm constant.
Thus when KE = ⅓ PE
We get PE = ¾ TE = ½ kz²
Hence we get ½ kx² = ¾ ½ kz²
We get x/z = √3/4
= 1.73 / 2
= .87
Thus we get x is 87 percent of the amplitude.

Answer

Answer

Given: The kinetic energy of a body executing S.H.M. is 1/3 of the potential energy.

To find: The displacement of the body is x percent of the amplitude.

Let's consider a body executing Simple Harmonic Motion (S.H.M.) with an amplitude A. The potential energy (PE) and kinetic energy (KE) of the body can be expressed as:

PE = (1/2)kx²
KE = (1/2)mv²

where k is the spring constant, x is the displacement from the mean position, m is the mass of the body, and v is the velocity of the body.

Since the kinetic energy is 1/3 of the potential energy, we can write:

(1/2)mv² = (1/3)(1/2)kx²
mv² = (1/3)kx²
v² = (kx²)/(3m)

Now, let's consider the relationship between displacement and velocity for a body executing S.H.M. We know that the maximum velocity occurs at the mean position (x = 0), and the velocity is zero at the extremes of the motion (x = ±A).

At the mean position (x = 0), the velocity is maximum and given by v_max = Aω, where ω is the angular frequency.

Substituting this into the equation v² = (kx²)/(3m), we get:

(Aω)² = (kx²)/(3m)
A²ω² = (kx²)/(3m)

Since ω = √(k/m) for S.H.M., we can rewrite the equation as:

A²(k/m) = (kx²)/(3m)
A² = (x²)/(3)

To find the displacement x as a percentage of the amplitude A, we can rearrange the equation:

x² = 3A²
x = √(3A²)
x = √3A

Since x is the displacement as a percentage of the amplitude A, we can calculate the percentage as:

Percentage = (x/A) * 100
Percentage = (√3A/A) * 100
Percentage = √3 * 100
Percentage = 173.2

Therefore, the displacement of the body is approximately 173.2% of the amplitude A.

However, none of the given options match this result. It seems there might be an error in the question or the options provided.

Top Courses for Class 11View all

Chemistry Class 11

Mathematics (Maths) Class 11

English Class 11

Career Awareness for Classes 11 & 12

Accountancy CUET UG Mock Test Series 2026

Top Courses for Class 11

Top Courses for Class 11

Related Content

Top Courses

Best Courses

Recommended Courses

CBSE Classes

Competitive Exams