Centre of Mass & Momentum PYQs Set 4 (PDF)

Set 4 of 4 · Questions 91-103 of 103

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Mere Teacher · Previous Year Questions JEE Main Physics · Centre of Mass & Momentum · Set 4 (Q91-103)
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91. $Consider two blocks A and B of masses m_{1} = 10 kg and m_{2} = 5 kg that are placed on a frictionless table. The block A moves with a constant speed v = 3 m / s towards the block B kept at rest. A spring with spring constant k = 3000 N / m is attached with the block B as shown in the figure. After the collision, suppose that the blocks A and B, along with the spring in constant compression state, move together, then the compression in the spring is, (Neglect the mass of the spring)$ (1) $0.3 m$ (2) $0.1 m$ (3) $0.4 m$ (4) $0.2 m$ 92. $Look at the drawing given in the figure which has been drawn with ink of uniform line-thickness. The mass of ink used to draw each of the two inner circles, and each of the two line segments is m. The mass of the ink used to draw the outer circle is 6m. The coordinates of the centres of the different parts are: outer circle (0, 0), left inner circle (-a, a), right inner circle (a, a), vertical line (0, 0) and horizontal line (0, -a). The y-coordinate of the centre of mass of the ink in this drawing is$ (1) $\frac{a}{10}$ (2) $\frac{a}{8}$ (3) $\frac{a}{12}$ (4) $\frac{a}{3}$ 93. $Two particles which are initially at rest move towards each other under the action of their mutual attraction. If their speeds are v and 2 v at any instant, then the speed of center of mass of the system is$ (1) $1.5 v$ (2) $2 v$ (3) $v$ (4) $zero$ 94. $A thin uniform rod of length λ and mass m is swinging freely about a horizontal axis passing through its end. Its maximum angular speed is ω. Its cenfre of mass rises to a maximum height of :$ (1) $\frac{1}{6} \frac{ℓ ω}{g}$ (2) $\frac{1}{2} \frac{ℓ ^{2} ω ^{2}}{g}$ (3) $\frac{1}{6} \frac{ℓ ^{2} ω ^{2}}{g}$ (4) $\frac{1}{3} \frac{ℓ ^{2} ω ^{2}}{g}$ 95. $Two persons of masses 55 kg and 65 kg respectively, are at the opposite ends of a boat. The length of the boat is 3.0 m and weighs 100 kg. The 55 kg man walks up to the 65 kg man and sits with him. If the boat is in still water the centre of mass of the system shifts by :$ (1) $3.0 m$ (2) $2.3 m$ (3) $zero$ (4) $0.75 m$ 96. $Four identical thin rods each of mass M and length λ, form a square frame. Moment of intertia of this frame about an axis through the centre of the square and perpendicular to its plane is$ (1) $\frac{4}{3} Mλ2$ (2) $\frac{2}{3} Mλ2$ (3) $\frac{13}{3} Mλ2$ (4) $\frac{1}{3} Mλ2$ 97. $A circular disc of radius R is removed from a bigger circular disc of radius 2R such that the circumferences of the discs coincide. The centre of mass of the new disc is αR distance from the centre of the bigger disc. The value of α is :$ (1) $1/3$ (2) $1/2$ (3) $1/6$ (4) $1/4$ 98. $A disc is rolling, the velocity of its centre of mass is v_{cm} . Which one will be correct?$ (1) $the velocity of highest point is 2 v_{cm} and point of contact is zero$ (2) $the velocity of highest point is v_{cm} and point of contact is v_{cm}$ (3) $the velocity of highest point is 2 v_{cm} and point of contact is v_{cm}$ (4) $the velocity of highest point is 2 v_{cm} and point of contact is 2 v_{cm} .$ 99. $As shown in figure. When a spherical cavity (centred at O) of radius 1 is cut out of a uniform sphere of radius R (centred at C), the centre of mass of remaining (shaded) part of sphere is at G, i.e on the surface of the cavity. R can be determined by the equation :$ (1) $(R^{2} - R + 1) (2 - R) = 1$ (2) $(R^{2} + R+- 1) (2-R) = 1$ (3) $(R^{2} - R - 1) (2 - R) = 1$ (4) $(R^{2} + R - 1) (2 - R) = 1$ 100. $A rigid ball of mass m strikes a rigid wall at 60º and gets reflected without loss of speed as shown in the figure below. The value of impulse imparted by the wall in the ball will be$ (1) $\frac{mV}{3}$ (2) $mV$ (3) $2mV$ (4) $\frac{mV}{2}$ 101. $A slender uniform rod of mass M and length l is pivoted at one end so that it can rotate in a vertical plane (see figure). There is negligible friction at the pivot. The free end is held vertically above the pivot and then released. the angular acceleration of the rod when it makes an angle θ with the vertical is$ (1) $\frac{2 g}{3 l} cos θ$ (2) $\frac{3 g}{2 l} s in θ$ (3) $\frac{2 g}{3 l} s in θ$ (4) $\frac{3 g}{2 l} cos θ$ 102. $A thin rod of length 'L' is lying along the x-axis with its ends at x = 0 and x = L. Its linear density (mass/length) varies with x as k (\frac{x}{L})^{n}, where n can be zero or any positive number. If the position x_{C M} of the centre of mass of the rod is plotted against 'n', which of the following graphs best approximates the dependence of x_{C M} on n ?$ (1) (2) (3) (4) 103. $A solid disc of radius 20cm and mass 10kg is rotating with an angular velocity of 600rpm, about an axis normal to its circular plane and passing through its centre of mass. The retarding torque required to bring the disc at rest in 10s is x π \times 1 0^{- 1} N m . The value of x is ____$