Question

2. Naruto's whack-a-smacker ninja-tool consists of a 3.42 kg copper ball of 9 cm diameter and a 4.1 kg steel ball of 10 cm diameter, rigidly joined by a 1.0 kg steel rod, 30 cm long. (a) Find the moment of inertia of this tool with respect to an axis that is perpendicular to the connecting rod and passes through its centre. Ignore the fact that the rod has finite diameter (here: 1.3 cm) and treat it as infinitely thin for the purpose of this problem. (4 marks) (b) Write the equation of motion for this tool, assuming that it is fixed to rotate around the same axis as in part (a). That axis of rotation is also set to be vertical. Assume that any friction can be neglected and that the only forces acting on the tool are due to gravity. (3 marks) (c) Assume that the tool is initially at rest at angle 0 =0. At to = 0 a constant external torque >0 is applied to it until time t1 > to, when it is removed again. Sketch the evolution of the angle of the tool, as a function of time, for times between ts <0 and t> (3 marks) Given: A uniform solid sphere of mass M and radius R has moment of inertia 1 =7MR around any axis passing through its centre. The moment of inertia of a rod of length L and mass M with respect to an axis passing through one of its ends, and perpendicular to its length, is 1 = 1; ML?

Answer 1

2. Naruto's whack-a-smacker ninja-tool consists of a 3.42 kg copper ball of 9 cm diameter and a 4.1 kg steel ball of 10 cm diameter, rigidly joined by a 1.0 kg steel rod, 30 cm long. a) m=liky L=30cm (a) Find the moment of inertia of this tool with respect to an axis that is perpendicular to the connecting rod and passes through its centre. Ignore the fact that the rod has finite diameter (here: 1.3 cm) and treat it as infinitely thin for the purpose of this problem. marks) (b) Write the equation of motion for this tool, assuming that it is fixed to rotate around the same axis as in part (a). That axis of rotation is also set to be vertical. Assume that any friction can be neglected and that the only forces acting on the tool are due to gravity. (3 marks) Steen) (c) 2 Assume that the tool is initially at rest at angle =0. At to =0 a constant external torque > 0 is applied to it until time > to, when it is removed again. Sketch the evolution of the angle of the tool, as a function of time, for times between ts <0 and (3 marks) A uniform solid sphere of mass M and radius R has moment of inertia / =% MR2 around any axis passing through its centre. The moment of inertia of a rod of length L and mass M with respect to an axis passing through one of its ends, and perpendicular to its length, is 1 = M12 Given: 2 Пев. + Гcs () m- 3.42 kg m=4-1kg r = 4.5cm r-5cm I: Irod +(I cut I ste y About central axis I=MRL 2 + + 2 ms 12 5 | x4.09) + 3x 3.42x60.045" + 1.42) S 12 2 1) I= 0.0075 + 0.003+ 0.08 +0.0041 1-014 = Id + 0.092 0.187 kg m² I SY K = 5. 42 radis? I- c) O: Wot+ça'ť It' x'= i = olrad) T => 0 = {( E) t² t(s)