dynamics hw

EXAM #2 (BONUS)

MEGR 3121

Max: 200 pts.

Due date: April 28, 2014 (in-class)

Total Problems: 4

 Show your coordinate systems clearly.

 Wherever relevant, clear and complete free body diagrams must be drawn.

 While using the work-energy equation, mention your datum clearly for the gravitational potential energy

 Each and every step involved in obtaining the final solution must be shown clearly. (Points may be deducted for following shortcuts)

 Units must be consistent.

 Points will be taken off as per grader’s discretion if your work is not legible

Student Name:

Student ID:

Student Signature:

For Grader’s Use:

Problem 1: /50

Problem 2: /50

Problem 3: /50

Problem 4: /50

Total: /200

2

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. [HINT: First, find P1 and P2]

 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.

[Justify mathematically]

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, y 2 = (4-x), where x, y are in cm w.r.t the XY system at O.

M

M

60°

60°

P1

P2

µs, µk

µs, µk

d

d

P

P P

(I) (II) (III)

H H

H

G G G

O

Y

X 15 cm

r

3

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. [You must show

that motion exists]. g = 10 m/s 2

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/s 2 )

MISCELLANEOUS

Length of a curve y = f(x) between x = a and x = b is given by the formula

C

40°

δ

R

d