Question

You have experienced a skiing accident and are trapped at the bottom of a ravine as shown in the figure below. Rescuers You During the accident, you cut your leg and it is bleeding continuously. Rescuers have arrived at the top of the ravine. The slope down to you is very steep; it is at 9 = 66.2° with respect to horizontal. The rescuers will require safety equipment to climb down to you. They have secured a post at the top of the ravine, d = 5.42 m from you, measured along the slope, and are going to send a medic down a rope tied to the post. Unfortunately, the cylindrical container holding the equipment for the rope climbing was accidentally dropped and has rolled down the slope to you. The rescuers do not have a replacement container and you don't have time for them to go back to headquarters and get another one, because of the bleeding in your leg. The rescuers call for you to roll the container up the hill so that it rolls all the way from you to them. The snow on the slope is such that the cylinder rolls without slipping, with no rolling friction. Assume the cylinder begins rolling as soon as you project it up the hill. Assume also that the mass in the cylinder is evenly distributed throughout. With what speed in m/s) must the center of mass of the cylinder begin its rolling motion at the bottom of the hill to reach the top? 9.859 Be sure to include the fact that the cylinder is rotating as it rolls up the hill. m/s

Answer 1

Sol: According to the question, The angle of the slope of reawine is, o=66.2 d dsimo -- = Now, d = 5.42 m. so, from the figure, height of the rearine, k= a smo Let M be the mass of the cylinder and R be the readines. So, moment of inertia of the cylinder is, I = MR? Let, zo be the speed of the center of mass fore the cylinder to reeach the lop. So, angulare velocity of the cylinder is, land of consercreation of Now, from the energy, we get Potential energy Kinetic energy at the bottom = at the top ore 2 M2 + I w? = Mgn ore 12 M 12 + 1 2 + 2 MR² x (12) Mgasime one, 22 Iasimo ore, ve ore, re = 8.04 So, the center of man of with a speed 8.04 mis at the treach the top. x 9.8 x 5.42 x 5m 66.272 ms. e of man of the cylinder must begin the cylinder reed 8.04 m/s at the bottom of the hill to