Question

Any assistance to get me started with (b), (C), (d) , and (e) would be appreciated. Thank you!

A system consisting of two pulleys, a spring, and four masses connected by strings is shown in the diagram below. Masses 1 and 2 rest on the side of a frictionless ramp at an angle θ from horizontal. mass 3 is suspended by a pulley, and mass 4 is hanging from a spring of spring constant k displaced a distance d from its equilibrium position. All strings and pulleys are of negligible mass. All four masses are at rest and are not accelerating a) Draw free-body diagrams for each of the four masses. Label each force with the type of interaction (e.g., weight W, normal force N, tension T, etc.) and a subscript indicating which object the force is acting upon and which object is exerting the force. For instance, the gravitational/weight force acting on mass 1 by the earth would be labeled Wie. b) Compute the magnitude of each force in terms of the following quantities: M1, M2, M3, Ma, g, θ, k, and c) Calculate the masses of blocks 3 and 4 in terms of Mi, M2, g, 0, k, and d d) The string connecting mass 1 and mass 2 is cut. What is the instantaneous acceleration of mass 3 immediately after this is done? e) After the string is cut, mass 1 slides a distance x down the ramp. What is the net external work done on the block in this time interval? 4

Answer 1

implies Spring constant k. displacement from the implies equilibrium position = delta x = d Choosing Tensions in the rope as shown in the figure #(1)FBD for m_1 #(2)FBD for m_2 #(3)FBD for m_3 #(4)FBD for m_4 T_1 = m_1g sin theta. (from. FBD 1) T_2 = m_2g sin theta + T_1 implies T_2 = m_2g sin theta + m_1g sin theta T_2 = (m_1 + m_2)g sin theta implies 2T_2 = m_3g implies T_2 = m_3g/2 Therefore(from FBD 3) Kd = T_2 + m_4 g From FBD 4 (d) T_2 = m_3g/2 Therefore T_2 = (m_1 + m_2)gsin theta. m_3g/2 = (m_1 + m_2)g sin theta. implies m_3 = 2(m_1 + m_2)sin theta. Kd = T_2 + m_4 g = (m_1 + m_2)g sin theta + m_4 g. implies Kd - (m_1 + m_2)g sin theta/g = m_4. (c) if T_2 rope is if then. FBD for m_3 is m_3g - T_2 = m_3a. m_3g - (m_1 + m_2)g sin theta = m_3a. a \r\n