Gravitational Field Intensity due to Ring at point on the axis
Let us consider a ring of mass M, having radius ‘a’; the gravitational field at a distance x along its axis is found as follows:
Consider a small length element along the circumferential length of the ring which has a mass ‘dm’; the field intensity due to this length element is given by,
dE = G(dm/r2)
The vertical components of the fields cancel each other due to the symmetry of the ring, and only horizontal components survive and add up to,
x is the distance of the point P from the center along the axis.
Q3: Why is the Gravitational Field at the Center of the Ring Zero?
A: At the center of the ring, the gravitational field due to each infinitesimal mass element of the ring cancels out symmetrically, leading to a net field intensity of zero.
Q4: How Does the Gravitational Field Vary Along the Axis?
A:
At the center of the ring (x = 0), the field is zero.
As x increases, the field initially increases, reaches a maximum, and then decreases asymptotically at large distances.
Q5: How Can This Concept Be Applied in Physics?
A: The concept of the gravitational field due to a ring is useful in astrophysics, planetary science, and engineering applications, including:
Understanding the gravitational effects of planetary rings.
Designing stable orbits in space missions.
Theoretical studies of symmetrical mass distributions.
FAQs on Gravitational Field Intensity Due to a Ring
Q1: Is the Gravitational Field Due to a Ring Uniform?
A: No, the gravitational field varies depending on the location of the point along the axis and is not uniform.
Q2: What Happens to the Gravitational Field as x → ∞?
A: As the distance x increases, the field intensity decreases and approximates the field due to a point mass MM at large distances.
Q3: What is the Direction of the Gravitational Field at a Point on the Axis?
A: The gravitational field is directed towards the center of the ring along the axis.
MCQs on Gravitational Field Intensity Due to a Ring
Q1: The gravitational field intensity at the center of a uniform ring is:
A) Maximum B) Zero C) Equal to GM/R² D) Infinite
Answer: B) Zero Explanation: The field intensity at the center cancels due to symmetry.
Q2: The gravitational field intensity due to a ring along its axis is maximum at:
A) The center of the ring B) Infinity C) A certain distance from the center D) On the ring itself
Answer: C) A certain distance from the center Explanation: The field first increases, reaches a maximum, and then decreases with distance.
Q3: The formula for the gravitational field intensity due to a ring is derived using:
A) Newton’s laws of motion B) Newton’s law of gravitation C) Einstein’s relativity D) Kepler’s laws
Answer: B) Newton’s law of gravitation Explanation: The field intensity is derived based on the inverse-square law of gravitation.
const questions = [
{ q: “What is the gravitational field intensity at the center of a uniform ring?”, options: [“Zero”, “Maximum”, “Infinity”, “Equal to mass of ring”], answer: 0, explanation: “At the center of a uniform ring, gravitational forces cancel out, making the field intensity zero.” },
{ q: “What is the formula for gravitational field intensity on the axis of a ring?”, options: [“Gm/x²”, “Gmz/(x² + r²)^(3/2)”, “Gm/r²”, “Gmz/(x² + r²)”], answer: 1, explanation: “The field intensity on the axis of a ring is given by Gmz/(x² + r²)^(3/2).” },
{ q: “How does gravitational field intensity change as we move along the axis of the ring?”, options: [“Increases then decreases”, “Remains constant”, “Continuously increases”, “Continuously decreases”], answer: 0, explanation: “The field first increases, reaches a maximum, and then decreases along the axis.” },
{ q: “Where is the maximum gravitational field intensity due to a ring located?”, options: [“At the center”, “At infinity”, “At a certain distance along the axis”, “Equally throughout the axis”], answer: 2, explanation: “The field is maximum at a certain point along the axis, not at the center or infinity.” },
{ q: “What happens to gravitational field intensity when the distance from the ring increases?”, options: [“Increases”, “Decreases”, “Remains the same”, “Becomes infinity”], answer: 1, explanation: “As distance increases, the field intensity decreases following inverse square law patterns.” },
{ q: “What is the gravitational field inside the ring along the plane of the ring?”, options: [“Zero”, “Maximum”, “Infinity”, “Same as at the axis”], answer: 0, explanation: “Inside the ring in its plane, the gravitational field intensity is zero due to symmetry.” },
{ q: “How does mass of the ring affect the gravitational field intensity?”, options: [“It does not affect”, “Field intensity is directly proportional to mass”, “Field intensity is inversely proportional to mass”, “Field intensity depends only on distance”], answer: 1, explanation: “The field intensity is directly proportional to the mass of the ring.” },
{ q: “What happens if the radius of the ring increases while keeping mass constant?”, options: [“Field intensity increases”, “Field intensity decreases”, “Field remains unchanged”, “Field becomes zero”], answer: 1, explanation: “Increasing the radius spreads the mass over a larger area, reducing field intensity.” },
{ q: “What is the nature of gravitational field intensity due to a ring?”, options: [“Vector quantity”, “Scalar quantity”, “Neither vector nor scalar”, “Both vector and scalar”], answer: 0, explanation: “Gravitational field intensity is a vector quantity because it has both magnitude and direction.” },
{ q: “How does gravitational field intensity change inside a solid ring?”, options: [“It remains constant”, “It increases”, “It decreases”, “It is always zero”], answer: 3, explanation: “Inside a uniform ring, the gravitational field intensity is always zero due to symmetry.” }
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