To release a small sphere of mass m hanging by a thread, the thread must first be pulled apart so that it makes a right angle with the vertical. Find: (a) the sphere's overall acceleration and the thread tension as a function of 0, the thread's angle of deviation from vertical.

A box with mass m is on a frictionless inclined ramp, as shown. The ramp has an angle of θ with respect to the horizontal.  The ramp is firmly attached to the inside of a car. The car is traveling to the right, and is slowing down. The car has a constant acceleration with magnitude a.  You observe that the box doesn’t slide up or down the ramp.     What is the angle of the ramp? Write your answer in variables, using only m, g, and a. This problem should be solved using the “recipe” taught in class. Focus Draw a free body diagram Identify the direction of the acceleration Write Newton's 2nd law (F=ma) Solve for unknowns  Suppose that the driver now presses harder on the brakes. Would the box slide up the ramp, or slide down the ramp? Or would it not slide at all? No calculation required, but very briefly (one or two sentences) justify your answer. Draw a simple diagram if it helps!

A sphere of mass M and radius R is not necessarily solid or hollow. It has moment of inertia I= cMR2. As shown in the figure, the sphere starts from rest and rolls without slipping down a ramp from height H. It then moves back up the other side, but now with no friction at all between the sphere and the ramp. What height does the sphere reach?

Derive the equations of motion using Lagrange's equations. (I, =mr? ) THE MOTION IS ON THE HORIZONTAL PLANE (NO GRAVITY) Trailer, mass M Cylinder, k1 mass m k2

Fixed Axis Rotation Example 4: (HW) Calculate the downward acceleration a of the 10-kg cylinder. The drum is a uniform cylinder, and friction at the pivot is negligible. Compare your answer with that obtained by ignoring the inertia effects of the drum. r = 250 mm 15 kg Ans. a = 0.769 m/s With Io = 0: a = 1.090 m/s2 8 kg 10 kg

4. A slender, rigid bar of mass m and length I is pinned to a massless collar as shown in the figure below. The collar and bar are initially stationary and slide on a frictionless rod. A force F is applied to the collar in the horizontal direction. Determine the acceleration of the collar at the instant that this force is applied. P Fight

Problem #2) The thin plate ABCD has mass 10kg and is held stationary in the position shown (p= 30°) by a wire BH and the two massless parallel links AF and DE. At t = 0sec, the wire is cut and the plate is released from rest; it immediately moves in a pure translation motion due to the system's geometric constraint by the links. Find the forces in each link immediately after the wire is cut. All dimensions are in millimeters. 150 A D -500 H B IC T 200

EXERCISE 3.40 Collar C slides relative to the curved rod at a constant speed u. The rotation rate about bearing axis AB is constant at Q. Determine the acceleration of the collar in terms of the angle locating the collar. Also derive expressions for the dynamic forces exerted on the collar by the rod and the tangential force required to hold u constant. Gravity may be assumed to be unimportant. (Hint: attach frame xyz to the curved rod, make the tangent point of AB and circle as its origin. Axis x is horizontal and rightward. y axis is vertical and downward.) 225