Toy A toy consists of two identical solid spheres connected by a string with negligible mass. The toy is thrown at an angle above the horizontal (not straight up) such that the string remains taut and both spheres are revolving counterclockwise in a vertical plane around the center of the string, as shown above. (a) Sketch the graphs of horizontal and vertical component of the velocity of the center of the string as a function of time, from the instant the spheres are released at time t = 0 until the instant the system returns to its initial height at time tr. Take the positive direction to be toward the right for the horizontal component and the positive direction to be upward for the vertical component.

Problem 1: A ball of mass m attached to a string is being swung around in a horizontal circle of radius R centered at the origin, at angular velocity wo. (Neglect gravity throughout this problem.) At time t = 0, the ball is on the +x-axis and the string breaks. Newton's First Law tells us that the now-free ball will move at constant velocity in a straight line parallel to the y-axis. However, the angular velocity of the ball is no longer constant. (a) Find an expression for the angular velocity as a function of time, w(t) = o(t), for this straight-line motion. Explicitly check your result for dimensional consistency and sensible limiting cases. (b) Assume instead that, after the string breaks, a purely angular force (i.e., zero radial component) acts on the ball in such a way as to keep its angular velocity constant at wo. Use the radial equation of motion in plane-polar coordinates to solve for the trajectory of the ball as a function of angle, p(p). Use Mathematica, Desmos, or some…

There are no numerical values assigned to the variables in this problem. A block of mass m1 is projected with a velocity vi at the top of the wedge. When the block reaches the horizontal surface its velocity vf to the right while the wedge moves with a velocity v2 of to the left. (a). What is the velocity of the once it reaches the horizontal surface? Your answer must be in terms of relevant variables. (b). What is the height h the wedge in terms of the relevant variables?

3 Given a wire frame in the shape of a right-angled triangle of vertical side a, of total mass m, with one side perpendicular to the z axis. The frame rotates about the z-axis with an angular velocity w. Find the L₂ component of the angular momentum of the frame.