a. What condition would need to be satisfied for the centre of mass of the dumbb

Question

a. What condition would need to be satisfied for the centre of mass of the dumbbell to be half way between the centres of the two masses? Explain your answer.

b. By what factor would you have to change the mass of the dumbbell to triple its kinetic energy, assuming a constant angular velocity and constant separation? Explain.

c. By what factor would you have to change the moment of inertia of the dumbbell to triple its kinetic energy, assuming a constant angular velocity? Explain.

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Explanation / Answer

a) The cnter of mass of the dumbel is given by XCM = (m1x1+ m2x2)/(m1+m2) For the center of mass to be at the half of the seperation the only possibile way is that the masses should be equal. At the center of mass m1(R/2)/(m1+m2) = m2(R/2)/(m1+m2) Therefore m1 = m2 b) The total kinetic energy of the dumbell is K = Ktran + Krot     = 0.5mv2 + 0.5I2 I is the moment of inertia of the dumbell = 0.5mR2 is the angular velocity = v/R K = 0.5mv2 + 0.5(0.5mR2)(v/R)2     = 0.75mv2 As K is proportional to the mass of the dumbell, the mass of the dumbell should be tripled in order to triple the kinetic energy c) K = 0.5(2I/R2)(/R)2 + 0.5I2     = 1.5I2 As K is proportional to the moment of inertia of the dumbell, the moment of inertia of the dumbell should be tripled in order to triple the kinetic energy     = 0.75mv2 As K is proportional to the mass of the dumbell, the mass of the dumbell should be tripled in order to triple the kinetic energy c) K = 0.5(2I/R2)(/R)2 + 0.5I2     = 1.5I2 As K is proportional to the moment of inertia of the dumbell, the moment of inertia of the dumbell should be tripled in order to triple the kinetic energy As K is proportional to the mass of the dumbell, the mass of the dumbell should be tripled in order to triple the kinetic energy c) K = 0.5(2I/R2)(/R)2 + 0.5I2     = 1.5I2 As K is proportional to the moment of inertia of the dumbell, the moment of inertia of the dumbell should be tripled in order to triple the kinetic energy As K is proportional to the moment of inertia of the dumbell, the moment of inertia of the dumbell should be tripled in order to triple the kinetic energy As K is proportional to the moment of inertia of the dumbell, the moment of inertia of the dumbell should be tripled in order to triple the kinetic energy

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