Question

Assignment:-GCA-CH07A work-KE Theory Pg12 SCALE-UP Table: Station: A block of mass M is launched up a rough plane that makes an angle θ with the horizontal by placing (not attached) it against a massless spring compressed by a distance d and releasing it from rest (see "Initial stage" in the figure at right). The block slides a distance L (>d) up the plane and comes off the plane with an unknown speed v (see "Final stage" in the figure at right). The spring constant is k and the coefficient of kinetic friction between the block and the plane isHe In this problem, you will use the W-KE theorem to derive an expression for v, in terms of given quantities and any other relevant constants Initial stage Final stage For M a). On the coordinate system provided at right, draw a complete FBD for Mas it is being pushed up the ramp by the spring. Why is it better to use a conventional coordinate system (horizontal-vertical axes) to solve this problem instead of a system in which the x-axis is parallel/anti parallel to the acceleration of the block? b). According to the coordinate system superimposed on the figure at top right, the block starts from the origin, i.e. at x-0 & y-0. Derive an expression for the block's y-coordinate when it comes off the plane at the top (y)? c). Derive an expression for the magnitude of the normal force exerted on the block by the plane. To do this, use the alternative coordinate system shown at right and apply Newton's 2nd law in the y* direction. For M P251

Answer 1

(a) When we choose the coordinate axis as vertical and horizontal axis, we can easily find the height of the block using trigonometery but if we take the coordinate axis as the parallel/antiparallel direction as x axis and perpendicular to the plane as y axis, then, in this case, we cannot find easily. So, we choose horizontal and vertical axis as the coordinate system. (b) From figure sin theta = y/L y = L sin theta cos theta = X/L rightwardsdoublearrow X = L cos theta (X, y) = (L sin theta, L cos theta) (c) (This is free body diagram when block is in contact with spring) Because acceleration along y* = a_y* = 0 net force along y* = 0 Sigma F_y* = ma_y* = 0 N - mg cos theta = 0 N = mg cos theta = normal force (d) kinetic friction is defined as f_k = -mu_k N (here, negative sign shows direction of acceleration is opposite to the motion of the block)