1) Jack and Jill are assigned the task of mowing Joe's yard. To add to their woes, the mowers given to them are identical, heavy, unmotorized and old-fashioned. Both are lazy and want to do minimum work (i.e. mechanical work) while mowing. Jack pushes the mower with minimum force P1 (as shown) while Jill pulls her mower with miniumum force P2. Both mowers are initially at rest. The ground obviously is rough (µs, µk).
Both mow till distance 'd'.
- Who did more (mechanical) work: Jack or Jill? Why? Justify by finding the expressions for work done by both Jack and Jill.
- Having covered a distance 'd', whose mower is moving faster: Jack's or Jill's? Justify by finding the expressions for final speeds of Jack and Jill's mowers.
2) A linear Hooke's spring has an unstretched length of 30 cm and a stiffness of 20 N/m. The spring is pivoted/fixed at point G (as shown) and is massless. A force P is applied to compress the spring from its initial position H to the pivot/fixed point G. Assume the spring follows the shape of the rods (as shown) at every instant. The force P is constant in magnitude and always tangential (during motion) to the rod/spring in each case. Neglect friction.
a) For which case the spring will store the most potential energy at G?
b) For which case the spring will store the most potential energy at H?
c) For which case the force P (needed to compress the spring to point G) will be the largest?
CASE I: Circular rod with radius, r = 14 cm
CASE II: Straight rod; length HG = 20 cm
CASE III: Parabolic rod, y2 = (4-x), where x, y are in cm w.r.t the XY system at O.
3) The two masses shown in figure have round frictionless massless pulleys. The inextensible cord connecting them is always taut. Given that F = 130 N; mA = mB = 30 kg, find the acceleration of the two blocks and tension in the cable when the system is released/pulled from rest. The floor is rough with µs = 0.2 and µk = 0.1.
4) What initial deformation (compression), δ, in the spring will make the mass leave the circular ramp at 40° (as shown)? The path is entirely frictionless. The mass initially starts off from rest. Note the mass and the spring are not rigidly attached to each other. (m = 1 kg, k = 200 N/m, R = 25 cm, d = 1 m, g = 10 m/s2)
