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MEKANIKA KLASIK I (2015/2016_1) The Kinematics of Rigid Body Motion 1. Prove that matrix multiplication is associative. Show that the product of two orthogonal matrices is also orthogonal?. 2. Show that the trace of a matrix is invariant under any similarity transformation. Show also that the antisymmetry property of a matrix is preserved under an orthogonal similarity transformation, while the hermitean property is invariant under any unitary similarity transformation. 3. The body set of axes can be related to the space set in terms of Euler`s angles by the following set of rotations: (1) Rotation about the x axis by an angle q (2) Rotation about the z’ axis by an angle y (3) Rotation about the old z axis by an angle f Show that this sequence leads to the same elements of the matrix of transformation as the sequence of rotations given in the book. [Hint: It is not necessary to carry out the explicit multiplication of the rotation matrices]. Small Oscilations 4. A mass particle moves in a constant vertical gravitational field along the curve defined by , where y is the vertical direction. Find the equation of motion for small oscillations about the position of equilibrium. 
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MEKANIKA KLASIK I (2015/2016_1) The Rigid Body Equation of Motion 1. If is an antisymmetric matrix associated with the coordinates of the ith mass point of a system, with elements show that the matrix of the inertia tensor can be written as PROBLEM 16.5 Intermediate Dynamics 2. Prove that for a general rigid body motion about a fixed point the time variation of the kinetic energy T is given by 3. (a) A bar of negligible weight and length l has equal mass points m at the two ends. The bar is made ti rotate uniformly about an axis passing through the center of the bar and making an angle q with the bar. Euler’s equations find components along the principal axes of the bar of the torque driving the bar. (b) From the fundamental torque equation find the components of the torque along axes fixed in space. Show that these components are consistent with those found in part (a). Small Oscilations 4. A 5atom linear molecule is simulated by a configuration of masses and ideal springs that looks like the following diagram: All force constants are equal. Find the eigenfrequencies and normal modes for longitudinal vibration. [Hint: transform the coordinates ηi = ζi defined by 
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