Chapter 9, Problem 9.19P

Four identical particles of mass mmm are mounted at equal intervals on a thin rod of length ℓ and mass M, with one mass at each end of the rod.     Part A If the system is rotated with angular velocity ω about an axis perpendicular to the rod through one of the end masses, determine the kinetic energy of the system.   Express your answer in terms of the variables m, M, l, and ωωomega.     PART B:If the system is rotated with angular velocity ω about an axis perpendicular to the rod through one of the end masses, determine the angular momentum of the system.

In the figure below about the point P and Ϝ find the sum of the moments of the two forces

Consider the Atwood machine, where two masses m1 = 12.6 kg and m2 = 4.2 kg are connected by an ideal wire that passes through a pulley of mass M = E of radius R = 0.15 m. The pulley is attached, but can rotate freely around its axis of symmetry. As shown in the Figure below, at the initial instant the mass m1 is at a height h= 3.29 m in relation to m2. Knowing that the system starts from rest, and that the magnitude of the linear velocity of m1 when the masses pass through the same vertical position is v=3.31 m/s, what is the mass M of the pulley? Here, assume the acceleration due to gravity as g=10 m/s², the pulley moment of inertia as I=MR2 and, finally, that the wire does not slide on the pulley. Choose one: a. 4,2 kg b. 14,7 kg c. 10,5 kg d. 6,3 kg e. 12,6 kg f. 16,8 kg g. 8,4 kg h. None of the other alternatives.

A physical pendulum composed of a solid sphere (radius R = 0.500m) is hanged from aceiling by string of length equal to radius. The moment of inertia of the solid sphere about itscenter of mass is Icm=(2/5)MR2 a. What is the total moment of inertia of the pendulum about its pivot at the ceiling? (useParallel Axis Theorem)b. What is the center of mass of the pendulum? (let the pivot point be the origin?c. What are the angular frequency, period, frequency of the system for small angles ofoscillation?

A solid cylinder has length L = 14 cm and radius R = 2 cm. The center of one face of the cylinder is located at the origin and the cylinder’s axis lies along the positive x-axis. The mass density of the cylinder varies along its length and is given by the function: ρ(x) = Ax2 + Bx + C, where A = 13.5 kg/m5, B = 5.2 kg/m4, and C = 14.2 kg/m3. Consider a thin slice of the cylinder that is a disk located at distance x from the origin and having thickness dx. Enter an expression for the mass of this disk in terms of the defined quantities and dx.  Integrate the expression you entered in part (a) and enter an expression for the mass of the whole cylinder in terms of the defined quantities. Calculate the mass of the cylinder, in grams. Enter an expression for the location of the cylinder’s center of mass along the x-axis, in terms of the defined quantities. Calculate the location of the cylinder’s center of mass along the x-axis, in centimeters.

Consider a flat rectangular plate of known mass, width and breadth with a negligible thickness that lies in the horizontal xy-plane. The plate is suspended from a thin piece of piano wire that is in the vertical orientation coincident to the z-axis and where the piano wire is attached to the center of the plate. When the plate is subjected to a torque whose vector is coincident to the z-axis, the plate rotates in the horizontal plane such that the rotation of the plate is modelled as θ=Csin⁡(ωnt+ϕ). The parameter information is: mass of plate M = 1.2 kilogram width of plate W = 0.040 meter breadth of plate B = 0.075 meter shear modulus of piano wire G = 79.3 gigaPascals diameter of piano wire D = 0.003 meter length of piano wire L = 0.120 meter amplitude of rotation C = 0.087267520415 radian phase lag of rotation ϕ = 1.565872597159 radian Using the supplied information and any appropriate assumptions and / or approximations to determine the following; 1) the mass moment of inertia I 2)…

For solid ojbject whose mass distribution does not allow for a simple determination of the center of mass by symmetry, the sums must be generalized to integrals where x and y are the coordinates of a small piece of an object that has mass dm. The integration is over the whole of the object. Consider a thin rod of length L, mass M, and cross-sectional area A. Let the origin of the coordinates be at the left end of the rod and the positive x-axis lie along the rod.      a. If the density p = M/V of the object is uniform, perform the integration described above to show that the x-coordinate of the center of mass of the rod is at its geometrical center.       b. If the density of the object varies linearly with x - that is, p = ax, where a is a positive constant - calculate the x-coordinate of the rod's center of mass.

Consider a physical system formed by a symmetrical rigid body with a circular profile of radius R. This rolls without sliding on a horizontal surface.   Which of the following statements is not correct?a. The velocity at the point of contact of the rigid body with the horizontal surface is zero. b. The speed of the center of mass of the rigid body is ωR c. The friction force of the rigid body with the ground is kinetic. d. The acceleration of the center of mass of the rigid body is αR

Consider a “round” rigid body with moment of inertia I = BMR2, where M is the body’s mass, R is the body’s radius, and B is a constant depending on the type of the body. The center of the “round” rigid body is attached to a spring of force constant k, and then the body is made to roll without slipping on a rough horizontal surface. Due to the spring, it is expected the body will oscillate by rolling back and forth from its resting position. A. Determine the angular frequency and the period for small oscillations of the round rigid body. Express your answer in terms of B. B. Among the four “round” rigid bodies shown at the table, for the same masses and radii, which among them will have the most number of cycles per second.