Consider again the situation given in Problem 1. (a) Find the center-of-mass of the system made up of m, and m, taking the origin to be at P. (b) Draw a free body force diagram with the all of the mass (m, + m,) concentrated at this center-of–mass position. (c) Find the torque about P due to (m, + m,) concentrated at this center of mass position. How does this result compare to (d) in Problem 1? [We will make use of this
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- A uniform circular disc of mass m = 5 kg and radius R=0.3 m may rotate about its fixed center of mass C. Starting from rest at t = 0, the disc is subjected to a constant pure couple Mo = 5 Newton-meters, as shown. At t= 0 a point A on the disc periphery is located as shown. Determine the acceleration aa of the point A when the disc has rotated through three and a half revolutions (express aa in Cartesians). M. Fixed A at t =0Find the center of mass of the following plane region with variable density. Describe the distribution of mass in the region. The upper half (y20) of the plate bounded by the ellipse x²+4y² = 4 with p(x,y)=1+y. The center of mass is ☐ (Type an ordered pair. Type an exact answer, using as needed.) Describe the distribution of mass in the region. Choose the correct answer below. O A. The density increases away from the x-axis. B. The density increases toward the x-axis. C. The density increases toward the y-axis. D. The density increases away from the y-axis.There are two boxes with same mass couple together by a massless rope. One of the masses is on a ramp raised to an angle of 42 degrees, the other mass is hanging off to the right as shown in the figure. In between the two masses, the rope sun over a frictionless, massless pulley. If the mass on the ramp starts the problem at the top of the inclined plane, and the system is given a push, so that it is moving with an initial velocity pointing down the ramp. Ignore wind resistance for this problem. a) If the coefficient of friction is 0.30, what is the magnitude and direction of the acceleration of the system? Use the coordinate system given in the figure. (a=-2.71362 m/s^2 along the -x direction) What is the angle required for the the acceleration of the system to equal zero? Explain you reasoning with a few sentences.
- 4.An experimental device imparts a force of magnitude F-34 lb to the front edge of the rim at A to simulate the effect of a slam dunk. Compute the moment of the force F about point O and about point B. The moments are positive if counterclockwise, negative if clockwise. Finally, locate a point Ċ from the base at O to the location on the ground where the force imparts zero moment. The distance d from point o to point C is positive if C is to the right of O, and negative if to the left. Assume a - 36 in., b - 28 in., h -12 in., H-10 ft, c-5, and d-9. b F Answers: Mo- MB- d= i i H lb-ft lb-ft ft(b) Two spheres, each of mass m, are connected by a rod. (The mass of the rod is so small that it can be ignored.) The X shows the position of the center of mass (cm) of the system. Three possible Mass m Mass m rotation axes, 1, 2, and 3, are shown. Rank the moments of inertia of cm the system for these three axes, from largest to smallest. Explain how you made your ranking. Axis Axis Axis # 1 #2 # 3 (c) Two spheres, one of mass m and one of mass 2m, are connected by a rod. (The mass of the rod is so small that it can be ignored.) The X shows the position of the center of mass (cm) of the system. Three Mass m Mass 2m possible rotation axes, 1, 2, and 3, are shown. Rank the moments of |cm inertia of the system for these three axes, from largest to smallest. Explain how you made your ranking. Аxis # 1 Аxis Axis # 3 # 2
- A uniform sphere of mass m and radius R rolls without slipping down a plane at an angle θ from thehorizontal. Show that the acceleration along the slope of the center of mass is aCM = (5/7)g sin θ and that the force of static friction needed is fs = (2/7)mg sin θ. What minimum coefcient of friction µs is needed to satisfythe conditions of the problem?Consider a ball rolling around in a circular path on the inner surface of a cone. The weight of the ball is shown by the vector W. Without friction, only one other force acts on the ball-a normal force, (a) Draw in the vector for the normal force. (The length of the vector depends on the next step, b.) (b) Using the parallelogram rule, show that the resultant of the two vectors is along the radial direction of the ball's circular path. (Yes, the normal is appreciably larger than the weight!)Consider the objects on the coordinate grid: a rod with m, = 5.25 kg, a right triangle with m, = 36.0 kg, and a square with m, = 4.45 kg. Calculate the center of gravity for the system. Assume the objects are uniform. (Give the coordinates of the center of gravity in m.) у (m) (9, 7) (-5, 5) (2,7) (8, 5) mo (-2, 2) (4, 1) x (m) x-coordinate y-coordinate m
- A projectile of mass m is launched with initial velocity Vo at angle (theta)from the +x direction. From the origin, the projectile is connected to a massless spring with force constant k, as shown. (1) Set-up the equations of motion of the projectile, both in the horizontal and vertical directions. Take note the presence of gravity and force by the spring (Hooke's law which is a vector).Problem 5: Two blocks are connected with a rope as shown. The surfaces and the pulley are frictionless. The pulley is a disk with radius r and mass m,. A force F is applied to Mass 1 as F m1 shown so that Mass 1 accelerates to the left and Mass 2 accelerates upward. Take the positive direction to the be the direction of accelerations of the masses. (That is, the positive direction for Mass 1 is to the left, and the positive direction for Mass 2 is upward.) mp m2 In the space below, show how you would find the acceleration of the masses. Start with equations from the equation sheet and show your steps to the point where you would calculate the results. You do not need to calculate any numerical result. Just show how you would solve the problem. Be sure your equations reflect the positive direction as described above. Max characters allowed 3000. 0/3000Show that, in general, a wrench can be replaced with two forces chosen in such a way that one force passes through a given point while the other force lies in a given plane.