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A particle, of unit mass moves along the x-axis under the influence of a potential, V (x) = x (x - 2 )2 . The particle is found to be in stable equilibrium at the point x = 2. Find out the time period of oscillation of the particle.Write answer upto two decimal places
Correct answer is '2.22'. Can you explain this answer?
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Verified Answer
A particle, of unit mass moves along the x-axis under the influence of...
The Potential is
V = x[x2 - 2x + 4]
V = x3 - 2x2 + 4x
For equilibrium
Given x0 = 2 then
= 2√2
V = x[x2 - 2x + 4]
V = x3 - 2x2 + 4x
For equilibrium
Given x0 = 2 then
= 2√22.22
Most Upvoted Answer
A particle, of unit mass moves along the x-axis under the influence of...
Given information:
- The potential function of the particle is V(x) = x(x - 2)^2.
- The particle is in stable equilibrium at x = 2.
Understanding the problem:
- The particle is in stable equilibrium at x = 2, which means it is in a position where it will not move unless an external force is applied.
- The potential function V(x) represents the energy of the particle at different positions along the x-axis.
Deriving the equation of motion:
- The equation of motion for a particle under the influence of a potential energy function is given by Newton's second law: F = -dV/dx, where F is the force acting on the particle.
- In this case, the force acting on the particle is F = -dV/dx = -d/dx(x(x - 2)^2).
- Taking the derivative, we get F = -2(x - 2)(x - 1).
Understanding the force:
- The force F = -2(x - 2)(x - 1) is a restoring force.
- At x = 2, the force is zero, indicating that the particle is in stable equilibrium.
- When the particle is displaced from x = 2, the force acts to restore it back to the equilibrium position.
Calculating the time period of oscillation:
- The time period of oscillation is the time taken for the particle to complete one full oscillation between two extreme positions.
- The time period can be determined by finding the time it takes for the particle to travel from one extreme position to another and back.
Finding the extreme positions:
- The extreme positions occur where the force F is maximum or minimum.
- The force F = -2(x - 2)(x - 1) is zero at x = 1 and x = 2.
- Therefore, the extreme positions are x = 1 and x = 2.
Calculating the time period:
- The time period T can be calculated using the formula T = 2π/ω, where ω is the angular frequency.
- The angular frequency ω is given by ω = √(k/m), where k is the force constant and m is the mass of the particle.
- In this case, the force constant k can be determined by evaluating the force F at x = 1 or x = 2.
- At x = 1, F = -2(1 - 2)(1 - 1) = 0, indicating that the particle is at rest.
- Therefore, we can conclude that the force constant k is positive.
- Since the mass of the particle is given as unit mass, we can assume m = 1.
Calculating the force constant:
- Let's calculate the force constant by evaluating the force F at x = 2.
- At x = 2, F = -2(2 - 2)(2 - 1) = 2.
- Therefore, the force constant k = 2.
Calculating the angular frequency:
- The angular frequency ω = √(k/m) = √(2/1) = √2.
Calculating the time period:
- The time period T
- The potential function of the particle is V(x) = x(x - 2)^2.
- The particle is in stable equilibrium at x = 2.
Understanding the problem:
- The particle is in stable equilibrium at x = 2, which means it is in a position where it will not move unless an external force is applied.
- The potential function V(x) represents the energy of the particle at different positions along the x-axis.
Deriving the equation of motion:
- The equation of motion for a particle under the influence of a potential energy function is given by Newton's second law: F = -dV/dx, where F is the force acting on the particle.
- In this case, the force acting on the particle is F = -dV/dx = -d/dx(x(x - 2)^2).
- Taking the derivative, we get F = -2(x - 2)(x - 1).
Understanding the force:
- The force F = -2(x - 2)(x - 1) is a restoring force.
- At x = 2, the force is zero, indicating that the particle is in stable equilibrium.
- When the particle is displaced from x = 2, the force acts to restore it back to the equilibrium position.
Calculating the time period of oscillation:
- The time period of oscillation is the time taken for the particle to complete one full oscillation between two extreme positions.
- The time period can be determined by finding the time it takes for the particle to travel from one extreme position to another and back.
Finding the extreme positions:
- The extreme positions occur where the force F is maximum or minimum.
- The force F = -2(x - 2)(x - 1) is zero at x = 1 and x = 2.
- Therefore, the extreme positions are x = 1 and x = 2.
Calculating the time period:
- The time period T can be calculated using the formula T = 2π/ω, where ω is the angular frequency.
- The angular frequency ω is given by ω = √(k/m), where k is the force constant and m is the mass of the particle.
- In this case, the force constant k can be determined by evaluating the force F at x = 1 or x = 2.
- At x = 1, F = -2(1 - 2)(1 - 1) = 0, indicating that the particle is at rest.
- Therefore, we can conclude that the force constant k is positive.
- Since the mass of the particle is given as unit mass, we can assume m = 1.
Calculating the force constant:
- Let's calculate the force constant by evaluating the force F at x = 2.
- At x = 2, F = -2(2 - 2)(2 - 1) = 2.
- Therefore, the force constant k = 2.
Calculating the angular frequency:
- The angular frequency ω = √(k/m) = √(2/1) = √2.
Calculating the time period:
- The time period T
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Community Answer
A particle, of unit mass moves along the x-axis under the influence of...
I think the answer will be 3.14 (pi). This question was asked on gate 2012. There was also the answer 3.14.
Solution:
V(x)= x(x-2)²
= x³ -4x²+4x
Now, Force F(x) = -dV/dx
= -3x² +8x-4
As the particle is in stable equilibrium at x=2, taking y=x-2,
F(y) = -3(y+2)² + 8(y+2) - 4
=-3y² -12y -12 +8y+16 -4
= -3y² -4y
Here the force component f(y) = -4y will cause simple harmonic motion. So, f(y) = -4y²
Comparing with f(y) = -ky², k= 4
So, T= 2π/w = 2π√(m/k) = 2π/√4 (as given, m=1)
or, T = π
Solution:
V(x)= x(x-2)²
= x³ -4x²+4x
Now, Force F(x) = -dV/dx
= -3x² +8x-4
As the particle is in stable equilibrium at x=2, taking y=x-2,
F(y) = -3(y+2)² + 8(y+2) - 4
=-3y² -12y -12 +8y+16 -4
= -3y² -4y
Here the force component f(y) = -4y will cause simple harmonic motion. So, f(y) = -4y²
Comparing with f(y) = -ky², k= 4
So, T= 2π/w = 2π√(m/k) = 2π/√4 (as given, m=1)
or, T = π
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A particle, of unit mass moves along the x-axis under the influence of a potential, V (x) = x (x - 2 )2. The particle is found to be in stable equilibrium at the point x = 2. Find out the time period of oscillation of the particle.Write answer upto two decimal placesCorrect answer is '2.22'. Can you explain this answer?
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A particle, of unit mass moves along the x-axis under the influence of a potential, V (x) = x (x - 2 )2. The particle is found to be in stable equilibrium at the point x = 2. Find out the time period of oscillation of the particle.Write answer upto two decimal placesCorrect answer is '2.22'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about A particle, of unit mass moves along the x-axis under the influence of a potential, V (x) = x (x - 2 )2. The particle is found to be in stable equilibrium at the point x = 2. Find out the time period of oscillation of the particle.Write answer upto two decimal placesCorrect answer is '2.22'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A particle, of unit mass moves along the x-axis under the influence of a potential, V (x) = x (x - 2 )2. The particle is found to be in stable equilibrium at the point x = 2. Find out the time period of oscillation of the particle.Write answer upto two decimal placesCorrect answer is '2.22'. Can you explain this answer?.
A particle, of unit mass moves along the x-axis under the influence of a potential, V (x) = x (x - 2 )2. The particle is found to be in stable equilibrium at the point x = 2. Find out the time period of oscillation of the particle.Write answer upto two decimal placesCorrect answer is '2.22'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about A particle, of unit mass moves along the x-axis under the influence of a potential, V (x) = x (x - 2 )2. The particle is found to be in stable equilibrium at the point x = 2. Find out the time period of oscillation of the particle.Write answer upto two decimal placesCorrect answer is '2.22'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A particle, of unit mass moves along the x-axis under the influence of a potential, V (x) = x (x - 2 )2. The particle is found to be in stable equilibrium at the point x = 2. Find out the time period of oscillation of the particle.Write answer upto two decimal placesCorrect answer is '2.22'. Can you explain this answer?.
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