3. A uniform disk of mass M and radius R compressed a spring (with spring constant k) a distance of xo from equilibrium. The spring is released and the disk begins to roll smoothly (there is no friction between the spring and the disk) up a ramp, as seen in (I) of the figure below. (1) +7 M, R (II) Max Height, H ++ k ¦¦M,R H }/ (a) What is the speed of the disk after it is released from the spring? (b) What is the maximum height, H, reached by the disk? Imagine now, as seen in (II), that the disk is reset to its initial position and the ramp is replaced with one that abruptly ends after a vertical height of H/2, where H is what you found in (b), and guides any rolling on it straight up. (c) The disk is once again released from the spring. What is the new maximum height it reaches (in mid-air)?

166. A physics experiment is set up using two pulleys, a string, and a weight as shown below. The larger pulley has a radius of 20 centimeters, and the smaller pulley has a radius of 5 centimeters. The distance between the centers of the pulleys is 51 centimeters. The string is pulled tightly across both pulleys so that AB is a common tangent of the pulleys. Find the length of string from A to B to the nearest tenth of a centimeter. B weight

A hoop of radius Rμ and mass mч and a solid cylinder of radius Rc and mass mc are released simultaneously at the top of a plane ramp of length L inclined at angle 0 above the horizontal plane. Which reaches the bottom of the plane first, and what is the speed of each? (solid cylinder)

In the figure here, a solid brass ball of mass 0.237 g will roll smoothly along a loop-the-loop track when released from rest along the straight section. The circular loop has radius R = 0.17 m, and the ball has radius r << R. (a) What is h if the ball is on the verge of leaving the track when it reaches the top of the loop? (b) If the ball is released at height h = 7R, what is the magnitude of the horizontal force component acting on the ball at point Q? (a) h= i (b) F= i R g < <

A spring with spring constant k and equilibrium length zero is attached tothe top of a frictionless hoop of radius R. The spring is stretched and connected to a bead of mass m at the bottom of the hoop. At t = 0 the bead is given an initial speed v0 and the bead moves up the hoop. Find the speed of the bead as a function of position on hoop v(θ).

A vertical spring stretches by 0.15 m to its new equilibrium length when a 3.0 N weight is attached. What is the spring constant?

A mass m = 77 kg slides on a frictionless track that has a drop, followed by a loop-the-loop with radius R = 19.9 m and finally a flat straight section at the same height as the center of the loop (19.9 m off the ground). Since the mass would not make it around the loop if released from the height of the top of the loop (do you know why?) it must be released above the top of the loop-the-loop height. (Assume the mass never leaves the smooth track at any point on its path.) ) What is the minimum speed the block must have at the top of the loop to make it around the loop-the-loop without leaving the track? m/s      What height above the ground must the mass begin to make it around the loop-the-loop? m      If the mass has just enough speed to make it around the loop without leaving the track, what will its speed be at the bottom of the loop? m/s      If the mass has just enough speed to make it around the loop without leaving the track, what is its speed at the…

Using the definition of dot product: Ax B= (A,B₂ - A₂By)i + (A₂Bx − AxB₂)j + (AxBy - Ay Bx)k Find the cross product Ax B the following vectors: (a) U A = 11 + 3) + 0k B = 3î + 1ĵ + 0k A = 11 - 3j+2k B = -31 + 1ĵ+ 0k (b) (d) A = 1î + 1ĵ + Ok B = 2î - 3j+0k A = 21-5j-1k B = 3î + 1ĵ+ 3k

Consider the rotational kinematic motion profile indicated below. t: 0 → 5 s = 0 deg/s , R O(0) = 0 deg = 0 deg/s Y(0) = 0 deg = 6 deg/s , Q Ф(0) = 0 deg O deg %3D t: 5 → 15 s P = 0 deg/s Ó = 6 deg/s Ý(t) = 3/cos(0(t)) deg/s Compute roll angle O at t = 15 s.

A ball of mass 2.3 kg, at one end of a string of length L=4.5 m, rotates in a vertical circle just fast enough to prevent the string from going slack at the top of the circle. Assuming mechanical energy is conserved, the speed of the ball (in m/s) at the bottom of the circle is: m a. 10.50 Ob. 8.13 Oc. 9.39 Od. 14.85 Oe. 6.64