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A particle is projected horizontally from the top of a tower with a velocity v0. If v be it's velocity at any instant, then the radius of curvature of the path of the particle at that instant is directly proportional to a) v^3 b) v^2 c) v d) 1/v?
Most Upvoted Answer
A particle is projected horizontally from the top of a tower with a ve...
Explanation:
When a particle is projected horizontally, it means its initial velocity has no vertical component. Therefore, the particle will only have horizontal motion and its vertical velocity will be zero at all times.
Radius of Curvature:
The radius of curvature represents the curvature of the path of the particle at any instant. It is the radius of the circle that best approximates the shape of the curve at that particular point.
Relationship between Velocity and Radius of Curvature:
The radius of curvature can be related to the velocity of the particle by examining the forces acting on the particle.
The only force acting on the particle in the horizontal direction is the initial velocity, which remains constant throughout the motion. Therefore, the velocity of the particle in the horizontal direction remains constant.
In the vertical direction, the only force acting on the particle is gravity. As a result, the velocity of the particle in the vertical direction increases linearly with time due to the acceleration due to gravity.
Centripetal Force:
As the particle moves horizontally, it experiences a centripetal force towards the center of curvature. This force is provided by the vertical component of the particle's velocity.
The centripetal force is given by the equation: Fc = mv^2 / r, where Fc is the centripetal force, m is the mass of the particle, v is the velocity, and r is the radius of curvature.
Relationship between Radius of Curvature and Velocity:
Using the equation for centripetal force, we can rearrange it to solve for the radius of curvature:
r = mv^2 / Fc
Since the force acting on the particle is the weight (mg) due to gravity, we can substitute Fc with mg:
r = mv^2 / mg
Simplifying the equation, we get:
r = v^2 / g
This equation shows that the radius of curvature is directly proportional to the square of the velocity (v^2) and inversely proportional to the acceleration due to gravity (g).
Answer:
Therefore, the radius of curvature of the path of the particle at any instant is directly proportional to v^2. Hence, the correct option is (b) v^2.
When a particle is projected horizontally, it means its initial velocity has no vertical component. Therefore, the particle will only have horizontal motion and its vertical velocity will be zero at all times.
Radius of Curvature:
The radius of curvature represents the curvature of the path of the particle at any instant. It is the radius of the circle that best approximates the shape of the curve at that particular point.
Relationship between Velocity and Radius of Curvature:
The radius of curvature can be related to the velocity of the particle by examining the forces acting on the particle.
The only force acting on the particle in the horizontal direction is the initial velocity, which remains constant throughout the motion. Therefore, the velocity of the particle in the horizontal direction remains constant.
In the vertical direction, the only force acting on the particle is gravity. As a result, the velocity of the particle in the vertical direction increases linearly with time due to the acceleration due to gravity.
Centripetal Force:
As the particle moves horizontally, it experiences a centripetal force towards the center of curvature. This force is provided by the vertical component of the particle's velocity.
The centripetal force is given by the equation: Fc = mv^2 / r, where Fc is the centripetal force, m is the mass of the particle, v is the velocity, and r is the radius of curvature.
Relationship between Radius of Curvature and Velocity:
Using the equation for centripetal force, we can rearrange it to solve for the radius of curvature:
r = mv^2 / Fc
Since the force acting on the particle is the weight (mg) due to gravity, we can substitute Fc with mg:
r = mv^2 / mg
Simplifying the equation, we get:
r = v^2 / g
This equation shows that the radius of curvature is directly proportional to the square of the velocity (v^2) and inversely proportional to the acceleration due to gravity (g).
Answer:
Therefore, the radius of curvature of the path of the particle at any instant is directly proportional to v^2. Hence, the correct option is (b) v^2.
Community Answer
A particle is projected horizontally from the top of a tower with a ve...
B) v^2
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A particle is projected horizontally from the top of a tower with a velocity v0. If v be it's velocity at any instant, then the radius of curvature of the path of the particle at that instant is directly proportional to a) v^3 b) v^2 c) v d) 1/v?
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A particle is projected horizontally from the top of a tower with a velocity v0. If v be it's velocity at any instant, then the radius of curvature of the path of the particle at that instant is directly proportional to a) v^3 b) v^2 c) v d) 1/v? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about A particle is projected horizontally from the top of a tower with a velocity v0. If v be it's velocity at any instant, then the radius of curvature of the path of the particle at that instant is directly proportional to a) v^3 b) v^2 c) v d) 1/v? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A particle is projected horizontally from the top of a tower with a velocity v0. If v be it's velocity at any instant, then the radius of curvature of the path of the particle at that instant is directly proportional to a) v^3 b) v^2 c) v d) 1/v?.
A particle is projected horizontally from the top of a tower with a velocity v0. If v be it's velocity at any instant, then the radius of curvature of the path of the particle at that instant is directly proportional to a) v^3 b) v^2 c) v d) 1/v? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about A particle is projected horizontally from the top of a tower with a velocity v0. If v be it's velocity at any instant, then the radius of curvature of the path of the particle at that instant is directly proportional to a) v^3 b) v^2 c) v d) 1/v? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A particle is projected horizontally from the top of a tower with a velocity v0. If v be it's velocity at any instant, then the radius of curvature of the path of the particle at that instant is directly proportional to a) v^3 b) v^2 c) v d) 1/v?.
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