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🔩 1. Helical Springs (Close-Coiled Compression Springs) Purpose : Energy absorption, vibration d…

April 14, 2025

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A table tennis ball has radius (3/2) × 10−2 m and mass (22/7) × 10−3 kg. It is slowly pushed down into a swimming pool to a depth of 𝑑 = 0.7 m below the water surface and then released from rest. It emerges from the water surface at speed 𝑣, without getting wet, and rises up to a height 𝐻. Which of the following option(s) is(are) correct? [Given: 𝜋 = 22/7, 𝑔 = 10 m s −2 , density of water = 1 × 103 kg m−3 , viscosity of water = 1 × 10−3 Pa-s.] (A) The work done in pushing the ball to the depth 𝑑 is 0.077 J. (B) If we neglect the viscous force in water, then the speed 𝑣 = 7 m/s. (C) If we neglect the viscous force in water, then the height 𝐻 = 1.4 m. (D) The ratio of the magnitudes of the net force excluding the viscous force to the maximum viscous force in water is 500/9.

Two uniform strings of mass per unit length u and 4u, and length L and 2L, respectively, are joined at point O, and tied at two fixed ends P and Q, as shown in the figure. The strings are under a uniform tension T. If we define the frequency v0 = 1/ 2L *[√T/u], which of the following statement(s) is(are) correct?

A thin uniform rod of length L and certain mass is kept on a frictionless horizontal table with a massless string of length L fixed to one end (top view is shown in the figure). The other end of the string is pivoted to a point O. If a horizontal impulse P is imparted to the rod at a distance x = L/n from the mid-point of the rod (see figure), then the rod and string revolve together around the point O, with the rod remaining aligned with the string. In such a case, the value of n is _____.

A glass beaker has a solid, plano-convex base of refractive index 1.60, as shown in the figure. The radius of curvature of the convex surface (SPU) is 9 cm, while the planar surface (STU) acts as a mirror. This beaker is filled with a liquid of refractive index � up to the level QPR. If the image of a point object O at a height of ℎ (OT in the figure) is formed onto itself, then, which of the following option(s) is(are) correct? (A) For � = 1.42, ℎ = 50 cm. (B) For � = 1.35, ℎ = 36 cm. (C) For � = 1.45, ℎ = 65 cm. (D) For � = 1.48, ℎ = 85 cm.

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causality paradox
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