Important Centre Of Mass Formulas For JEE
The concept of the center of mass is a fundamental principle in physics and engineering, and it plays a crucial role in understanding the distribution of mass within objects. When preparing for the Joint Entrance Examination (JEE), mastering the center of mass formulas is essential, as it is a topic that appears across various subjects, including mechanics and dynamics. In this article, we will explore some of the important centers of mass formulas that can greatly assist JEE aspirants in their preparation.
Center of Mass Overview:
The center of mass of an object is the point at which the entire mass of the object is considered to be concentrated. It is a point that represents the average position of the mass distribution. Understanding center of mass is vital as it helps solve complex problems related to motion, collisions, equilibrium, and more.
Center of Mass Formula Table :
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Centre of Mass Formula |
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Description |
Formula |
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Centre of mass of system on a line |
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Centre of mass of system on a 2D plane |
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Centre of mass of rectangular plate |
rx = B / 2 |
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Centre of mass of triangular plate |
rc = h / 3 |
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Centre of mass of semi-circular ring |
ry = 2R / π, rx = O |
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Centre of mass of semi-circular disc |
ry = 4R / (3π), rx = O |
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Centre of mass of hemispherical shell |
ry = R / 2, rx = O |
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Centre of mass of solid hemisphere |
ry = 3R / (8), rx = O |
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Centre of mass of circular cone |
ry = h / 4 |
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Centre of mass of hollow circular cone |
ry = h / 3 |
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Circular Motion Formula |
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Description |
Formula |
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Average angular velocity |
ωavg = (θ2 - θ1) / (t2 - t1) |
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Average angular acceleration |
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Tangential acceleration |
at = dV / dt |
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Centripetal acceleration |
ac = v² / r |
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Normal reaction on concave bridge |
N = mg cosθ + (mv²) / r |
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Normal reaction on convex bridge |
N = mg cosθ - (mv²) / r |
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Safe velocity on level road |
vsafe ≤ √(μ * g * r) |
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Banking angle |
tanθ = v² / (g * r) |
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Centrifugal force |
f = m * ω² * r |
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Conical pendulum |
T = 2π * √((Lcosθ) / g) |
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De Broglie Wavelength Formula |
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Description |
Formula |
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De Broglie wavelength |
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Radius of electron in atoms |
rn = n² / (Z * a₀) |
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Speed of electron in atoms |
vn = Z / n * v₀ |
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Energy in nth orbit |
En = E₁ * Z² / n² |
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Wavelength corresponding to lines |
1/λ = R * (1/n₁² - 1/n₂²) |
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Min. wavelength for x-rays |
λmin = hc / eV₀ |
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Radius of nucleus |
R = R₀ * A^(1/3) |
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Number of nuclei in decay |
N = N₀ * e^(-λt) |
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Half-life of radioactive sample |
T(1/2) = 0.693 / λ |
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Average life |
Tav = T(1/2) / 0.693 |
Mastering center of mass formulas is crucial for JEE preparation, as it can help solve complex problems in mechanics and dynamics. These formulas provide insights into the distribution of mass within various objects, allowing students to analyze motion, equilibrium, and other physical phenomena. By understanding and applying these formulas effectively, JEE aspirants can enhance their problem-solving skills and perform well in the examination. It's important to practice a variety of problems related to center of mass to gain confidence and proficiency in using these formulas during the JEE.
Important Centre Of Mass Formulas For JEE FAQs
Q1: What is the center of mass?
Ans: The center of mass is a point within an object or system where the entire mass can be considered to be concentrated. It's the point around which an object will balance in any orientation.
Q2: Why are center of mass formulas important for JEE preparation?
Ans. Center of mass formulas are essential for understanding complex physics problems related to motion, equilibrium, collisions, and more. They appear across various topics in JEE, making them crucial for problem-solving.
Q3. What is the significance of the center of mass in different shapes?
Ans. The center of mass in different shapes helps us understand the balance and stability of objects. For example, in a rectangular plate, it's at the intersection of diagonals, and in a triangular plate, it's one-third the height from the base.
Q4. How do center of mass formulas apply to circular objects?
Ans. Center of mass formulas for circular objects, such as semi-circular rings or discs, provide insights into their balancing points. These formulas aid in solving problems related to rotational motion and equilibrium.
Q5. Are there scenarios where the center of mass calculations become more complex?
Ans. Yes, scenarios like calculating the center of mass for a circular cone or a hollow circular cone require understanding the distribution of mass within the shape. The formulas provide insights into their balance points.
