Physics 312, Winter 2007, Practice Final Time: Two hours Answer one of Question 1 or Question 2 plus one of Question 3 or Question 4 plus one of Question 5 or Question 6. Each question carries equal weight. Partial credit for incomplete answers will be given. Explain your answers in detail. ANSWER: 1. Consider a solid, uniform, cone of mass m, base radius a, and height h. (a) Calculate the inertia tensor of the cone in the co-ordinate system shown below. (b) What are the principal axes of the cone in this co-ordinate system? (c) If the cone spins about the z-axis with an angular speed ω what is its rotational kinetic energy? OR: 2. A uniform block of mass m and dimensions a by 2a by 3a spins about a long diagonal with angular speed ω. Using a co-ordinate system whose origin is the centre of the block, and whose axes are fixed within the block ( body system ): (a) Compute the inertia tensor for the block; (b) Write down Euler s equations and use them to find the torque N that (in this co-ordinate system) must be exerted on the block if its angular velocity vector is to be constant in magnitude and direction while it spins in the manner described.
AND: 3. Consider a rigid body with an axis of symmetry. That is to say, a body whose principal moments of inertia obey: I 1 = I 2 = I; I 3 = I S I. (1) This problem concerns the behaviour of the rotational velocity vector ω in the co-ordinate system defined by the three principal axes of the body ( body coordinate system). We will discuss the behaviour of ω in the case that there is no torque on the body. (a) Write down Euler s equations of motion for this body and hence show that ω 3, the component of ω along the axis of symmetry, is a constant, if N = 0. (b) By considering the other two Euler equations for this body show that: ω 1 + ω 2 Ω = 0 ω 2 ω 1 Ω = 0, with Ω = ω 3 ( IS I 1 ). (c) Deduce that ω 1 can be chosen to be proportional to cos(ωt) and sketch the path that the vector ω traces out in this co-ordinate system. (d) The Earth is an oblate spheroid with I S /I = 1.00327. The Earth s axis of rotation is inclined to its symmetry axis at an angle of approximately 0.2 seconds of arc. What is angular frequency with which the Earth s rotation axis precesses about its symmetry axis?
OR: 4. (a) Draw a sketch to indicate how the Euler angles φ, θ, and ψ are used to relate the space co-ordinate system {ˆx, ŷ, ẑ} to the body co-ordinate system {ê 1, ê 2, ê 3 }. (b) Describe the three rotations that take us from the space system to the body system and write down the three matrices for these rotations. (c) Now consider a body with an axis of symmetry, i.e. a body obeying Eq. (1) which is not being acted upon by any external torques. If the angular momentum vector L is parallel to ẑ, explain why, for such a body, L may be written as: L = L sin θê 2 + L cosθê 3, with θ the Euler angle discussed above. (d) If the angle between the axis of rotation and ê 3 is α use the relationship between L and the angular velocity vector in the body co-ordinate system to deduce that: tanθ = I I S tan α. (2) (e) The Earth is an oblate spheroid with I S /I = 1.00327. The Earth s axis of rotation is inclined to its symmetry axis at an angle of approximately 0.2 seconds of arc. What is the angle between the Earth s angular momentum vector and its axis of rotation? (Note: The question asks for the angle between L and ω, not the angle between L and the symmetry axis.)
AND: 5. A particle is free to slide, under the influence of gravity, along a smooth cycloidal trough whose surface is given by the parametric equations: x(θ) = a (2θ + sin(2θ)), 4 y(θ) = 0, z(θ) = a (1 cos(2θ)), 4 where π θ π and a is a constant. (See diagram.) (a) State Hamilton s principle (the principle of least action). (b) Only one generalized co-ordinate is needed here. Why? What is it? (c) What is the Lagrangian in terms of this co-ordinate? (d) Find the equation of motion of the particle. (e) Extra credit: By defining s = a sin θ demonstrate that the particle executes simple harmonic motion with frequency g/a.
OR: 6. A particle slides (frictionlessly) on a smooth inclined plane whose inclination θ is increasing at a constant rate ω, with θ = 0 at t = 0. (See figure below.) (a) State Hamilton s principle (the principle of least action). (b) Only one generalized co-ordinate is needed here. Why? (c) Choose the co-ordinate to be the distance of the particle from the pivot point of the plane, r(t), and derive the Lagrangian L of the particle in terms of r(t), ṙ(t), ω, and t. (d) Hence obtain the equation of motion for the co-ordinate r(t). (e) Assuming that the particle is initially at rest, and located a distance x 0 from the plane s pivot point (see figure), show that: r(t) = x 0 cosh(ωt) + g 2ω2(sin(ωt) sinh(ωt)). solves the equation of motion derived in part (d) and obeys the initial conditions.