Centre of mass PYQs: JEE Mains Physics

Showing 1–42 of 42 questions

1. 1. The position of center of mass of three masses \(2\) kg, \(3\) kg and \(15\) kg placed with respect to mid point (\(p\)) of normal bisector, as shown in the figure is _______.
   2026 Hard
2. 2. A rod of length 5 L is bent right angle keeping one side length as 2 L.
   
   The position of the centre of mass of the system: (Consider \(\mathrm{L}=10 \mathrm{~cm}\)) 2025 Medium
3. 3. Consider a circular disc of radius 20 cm with centre located at the origin. A circular hole of radius 5 cm is cut from this disc in such a way that the edge of the hole touches the edge of the disc. The distance of centre of mass of residual or remaining disc from the origin will be ________. 2025 Medium
4. 4. The center of mass of a thin rectangular plate (fig - x ) with sides of length \(a\) and \(b\), whose mass per unit area \((\sigma)\) varies as \(\sigma=\frac{\sigma_0 x}{a b}\) (where \(\sigma_0\) is a constant), would be \(\qquad\)
   
   2025 Medium
5. 5. A uniform thin metal plate of mass \(10 \mathrm{~kg}\) with dimensions is shown. The ratio of \(x\) and \(y\) coordinates of center of mass of plate in \(\frac{n}{9}\). The value of \(n\) is _______.
   2024 Hard
6. 6. In a system two particles of masses \(m_1=3 \mathrm{~kg}\) and \(\mathrm{m}_2=2 \mathrm{~kg}\) are placed at certain distance from each other. The particle of mass \(m_1\) is moved towards the center of mass of the system through a distance \(2 \mathrm{~cm}\). In order to keep the center of mass of the system at the original position, the particle of mass \(\mathrm{m}_2\) should move towards the center of mass by the distance _______\(\mathrm{cm}\). 2024 Hard
7. 7. The identical spheres each of mass \(2 \mathrm{M}\) are placed at the corners of a right angled triangle with mutually perpendicular sides equal to \(4 \mathrm{~m}\) each. Taking point of intersection of these two sides as origin, the magnitude of position vector of the centre of mass of the system is \(\frac{4 \sqrt{2}}{x}\), where the value of \(x\) is _______. 2024 Hard
8. 8. The distance of centre of mass from end \(A\) of a one dimensional rod \(( AB )\) having mass density \(\rho=\rho_{0}\left(1-\frac{ x ^{2}}{ L ^{2}}\right)\,kg / m\) and length \(L\) (in meter) is \(\frac{3 L }{\alpha} m\). The value of \(\alpha\) is \(\ldots \ldots \ldots . .\) (where \(x\) is the distance form end \(A\) ) 2022 Hard
9. 9. Three identical spheres each of mass \(M\) are placed at the corners of a right angled triangle with mutually perpendicular sides equal to \(3\,m\) each. Taking point of intersection of mutually perpendicular sides as origin, the magnitude of position vector of centre of mass of the system will be \(\sqrt{x} m\). The value of \(x\) is 2022 Medium
10. 10. A circular hole of radius \(\left(\frac{ a }{2}\right)\) is cut out of a circular disc of radius \('a'\) as shown in figure. The centroid of the remaining circular portion with respect to point \('O'\) will be :
    2021 Hard
11. 11. The disc of mass \(M\) with uniform surface mass density \(\sigma\) is shown in the figure. The centre of mass of the quarter disc (the shaded area) is at the position \(\frac{x}{3} \frac{a}{\pi}, \frac{x}{3} \frac{a}{\pi}\) where \(x\) is ....... . (Round off to the Nearest Integer) \([ a\) is an area as shown in the figure \(]\)
    2021 Medium
12. 12. The position of the centre of mass of a uniform semi-circular wire of radius \('R'\) placed in \(x-y\) plane with its centre at the origin and the line joining its ends as \(x\)-axis is given by \(\left(0, \frac{x R}{\pi}\right)\). Then, the value of \(|x|\) is ...... . 2021 Medium
13. 13. A rod of length \(L\) has non-uniform linear mass density given by \(\rho(\mathrm{x})=\mathrm{a}+\mathrm{b}\left(\frac{\mathrm{x}}{\mathrm{L}}\right)^{2},\) where \(a\) and \(\mathrm{b}\) are constants and \(0 \leq \mathrm{x} \leq \mathrm{L}\). The value of \(\mathrm{x}\) for the centre of mass of the rod is at 2020 Hard
14. 14. A rod of length \(L\) has non-uniform linear mass density given by \(\rho(\mathrm{x})=\mathrm{a}+\mathrm{b}\left(\frac{\mathrm{x}}{\mathrm{L}}\right)^{2},\) where \(a\) and \(\mathrm{b}\) are constants and \(0 \leq \mathrm{x} \leq \mathrm{L}\). The value of \(\mathrm{x}\) for the centre of mass of the rod is at 2020 Hard
15. 15. A square shaped hole of side \(l=\frac{a}{2}\) is carved out at a distance \(d =\frac{ a }{2}\) from the centre \('O'\) of a uniform circular disk of radius \(a\). If the distance of the centre of mass of the remaining portion from \(O\) is \(-\frac{a}{X},\) value of \(X\) (to the nearest integer) is.......
    2020 Hard
16. 16. A square shaped hole of side \(l=\frac{a}{2}\) is carved out at a distance \(d =\frac{ a }{2}\) from the centre \('O'\) of a uniform circular disk of radius \(a\). If the distance of the centre of mass of the remaining portion from \(O\) is \(-\frac{a}{X},\) value of \(X\) (to the nearest integer) is.......
    2020 Hard
17. 17. As shown in figure, when a spherical cavity (centred at \(\mathrm{O})\) of radius \(1\) is cut out of a uniform sphere of radius \(\mathrm{R} \text { (centred at } \mathrm{C}),\) the centre of mass of remaining (shaded) part of sphere is at \(G\), i.e, on the surface of the cavity. \(\mathrm{R}\) can be detemined by the equation
    2020 Hard
18. 18. As shown in figure, when a spherical cavity (centred at \(\mathrm{O})\) of radius \(1\) is cut out of a uniform sphere of radius \(\mathrm{R} \text { (centred at } \mathrm{C}),\) the centre of mass of remaining (shaded) part of sphere is at \(G\), i.e, on the surface of the cavity. \(\mathrm{R}\) can be detemined by the equation
    2020 Hard
19. 19. The centre of mass of a solid hemisphere of radius \(8\, cm\) is \(X \,cm\) from the centre of the flat surface. Then value of \(x\) is\(......\) 2020 Medium
20. 20. The centre of mass of a solid hemisphere of radius \(8\, cm\) is \(X \,cm\) from the centre of the flat surface. Then value of \(x\) is\(......\) 2020 Medium
21. 21. The coordinates of centre of mass of a uniform flag shaped lamina (thin flat plate) of mass \(4\;kg\). (The coordinates of the same are shown in figure) are
    2020 Medium
22. 22. The coordinates of centre of mass of a uniform flag shaped lamina (thin flat plate) of mass \(4\;kg\). (The coordinates of the same are shown in figure) are
    2020 Medium
23. 23. Three point particles of masses \(1.0\; \mathrm{kg} .1 .5 \;\mathrm{kg}\) and \(2.5\; kg\) are placed at three comers of a right angle triangle of sides \(4.0\; \mathrm{cm}, 3.0 \;\mathrm{cm}\) and \(5.0\; \mathrm{cm}\) as shown in the figure. The center of mass of the system is at a point
    2020 Medium
24. 24. Three point particles of masses \(1.0\; \mathrm{kg} .1 .5 \;\mathrm{kg}\) and \(2.5\; kg\) are placed at three comers of a right angle triangle of sides \(4.0\; \mathrm{cm}, 3.0 \;\mathrm{cm}\) and \(5.0\; \mathrm{cm}\) as shown in the figure. The center of mass of the system is at a point
    2020 Medium
25. 25. A piece of wood of mass \(0.03\, kg\) is dropped from the top of a \(100\, m\) height building. At the same time, a bullet of mass \(0.02\, kg\) is fired vertically upward, with a velocity \(100\, ms^{- 1}\), from the ground. The bullet gets embedded in the wood. Then the maximum height to which the combined system reaches above the top of the building before falling below is ........ \(m\). \((g = 10\, ms^{-2})\) 2019 Hard
26. 26. A piece of wood of mass \(0.03\, kg\) is dropped from the top of a \(100\, m\) height building. At the same time, a bullet of mass \(0.02\, kg\) is fired vertically upward, with a velocity \(100\, ms^{- 1}\), from the ground. The bullet gets embedded in the wood. Then the maximum height to which the combined system reaches above the top of the building before falling below is ........ \(m\). \((g = 10\, ms^{-2})\) 2019 Hard
27. 27. A uniform rectangular thin sheet \(ABCD\) of mass \(M\) has length \(a\) and breadth \(b\), as shown in the figure. If the shaded portion \(HBGO\) is cut off, the coordinates of the centre of mass of the remaining portion will be
    2019 Hard
28. 28. A uniform rectangular thin sheet \(ABCD\) of mass \(M\) has length \(a\) and breadth \(b\), as shown in the figure. If the shaded portion \(HBGO\) is cut off, the coordinates of the centre of mass of the remaining portion will be
    2019 Hard
29. 29. Four particles \(A, B, C\) and \(D\) with masses \(m_A=m, m_B=2m, m_C=3m\) and \(m_D=4m\) are at the corners of a square. They have accelerations of equal magnitude with directions as shown. The acceleration of the centre of mass of the particles is
    2019 Hard
30. 30. Four particles \(A, B, C\) and \(D\) with masses \(m_A=m, m_B=2m, m_C=3m\) and \(m_D=4m\) are at the corners of a square. They have accelerations of equal magnitude with directions as shown. The acceleration of the centre of mass of the particles is
    2019 Hard
31. 31. The position vector of the centre of mass \(\vec r\, cm\) of an asymmetric uniform bar of negligible area of cross-section as shown in figure is
    2019 Medium
32. 32. The position vector of the centre of mass \(\vec r\, cm\) of an asymmetric uniform bar of negligible area of cross-section as shown in figure is
    2019 Medium
33. 33. Three particles of masses \(50\, g\), \(100\, g\) and \(150\, g\) are placed at the vertices of an equilateral triangle of side \(1\, m\) (as shown in the figure). The \((x, y)\) coordinates of the centre of mass will be
    2019 Hard
34. 34. Three particles of masses \(50\, g\), \(100\, g\) and \(150\, g\) are placed at the vertices of an equilateral triangle of side \(1\, m\) (as shown in the figure). The \((x, y)\) coordinates of the centre of mass will be
    2019 Hard
35. 35. In the figure shown \(ABC\) is a uniform wire . If centre of mass of wire lies vertically below point \(A\) , then \(\frac{{BC}}{{AB}}\) is close to 
    2016 Hard
36. 36. In the figure shown \(ABC\) is a uniform wire . If centre of mass of wire lies vertically below point \(A\) , then \(\frac{{BC}}{{AB}}\) is close to 
    2016 Hard
37. 37. A uniform thin rod \(AB\) of length \(L\) has linear mass density \(\mu \left( x \right) = a + \frac{{bx}}{L}\) , where \(x\) is measured from \(A\). If the \(CM\) of the rod lies at a distance of \(\left( {\frac{7}{12}} \right)L\) from \(A\), then \(a\) and \(b\) are related as 2015 Hard
38. 38. A uniform thin rod \(AB\) of length \(L\) has linear mass density \(\mu \left( x \right) = a + \frac{{bx}}{L}\) , where \(x\) is measured from \(A\). If the \(CM\) of the rod lies at a distance of \(\left( {\frac{7}{12}} \right)L\) from \(A\), then \(a\) and \(b\) are related as 2015 Hard
39. 39. Distance of the centre of mass of a solid uniform cone from its vertex is \(z_0\) . If the radius of its base is \(R\) and its height is \(h\) then \(z_0\) is equal to 2015 Hard
40. 40. Distance of the centre of mass of a solid uniform cone from its vertex is \(z_0\) . If the radius of its base is \(R\) and its height is \(h\) then \(z_0\) is equal to 2015 Hard
41. 41. A thin bar of length \(L\) has a mass per unit length \(\lambda \), that increases linearly with distance from one end. If its total mass is \(M\) and its mass per unit length at the lighter end is \(\lambda_0\), then the distance of the centre of mass from the lighter end is 2014 Hard
42. 42. A thin bar of length \(L\) has a mass per unit length \(\lambda \), that increases linearly with distance from one end. If its total mass is \(M\) and its mass per unit length at the lighter end is \(\lambda_0\), then the distance of the centre of mass from the lighter end is 2014 Hard

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