I need help in dynamics hw ( mostly physics)

MEGR 3121, Dynamics Systems – I

HW # 6, Due: April 16, 2014

Max: 300 + 15 Free body diagrams must be drawn, wherever relevant Clearly indicate the coordinate systems you are using to solve the problem.

Wherever appropriate, make use of F = ma and/or Work-Energy equation. (g = 10 m/s 2 or 32.2

ft/s 2 )

1) The 0.5 kg pellet is pushed against the spring at A and released from rest. Neglecting

friction, determine the smallest deflection of the spring for which the pellet will travel around

the loop ABCDEB without losing contact with it. Note the mass is not rigidly attached to the

spring.

2) A block of mass 5 kg has a speed of 4 m/s at point A and is continuously subjected to the

force P of 20 N as shown throughout its motion from A to B. AB is a rough surface with

coefficient of friction = 0.2. Compute the speed of the block at point B using Work-Energy

principle. The distance between A and B is 2 meters. [Note, be careful of the sign of friction

work]

3) The 2-kg collar is released from rest at A as shown and slides down the inclined fixed rod

in the vertical plane. The coefficient of kinetic friction is 0.4. Calculate (a) the velocity of the

collar as it strikes the spring and (b) the maximum deflection of the spring

400 N/m

m = 0.5 kg

5 cm

5 kg 15°

P=20 N

A B

VA = 4 m/s

2 m, rough, µ = 0.2

4) The small slider of mass m is released from rest while in position A and then slides along

the vertical-plane track. The track is smooth everywhere and gets rough (coefficient of

kinetic friction μk) from point D onward. Determine (a) the normal force NB exerted by the

track on the slider when it is at B, and (b) the distance s traveled along the incline past point

D before the slider stops.

5) The force P = 40 N is applied to the system, which is initially at rest. Determine the speeds

of A and B after A has moved 0.4 meters. Neglect friction.

6) The spring, as shown in the figure, has a stiffness of 3 N/m and is unstretched when

the slider is at A. If the speed vA is such that the speed of the 0.4 kg slider approaches

zero at C, determine (a) the speed of the slider at point B, and the (b) the normal force at

point B. The equation of the elliptical track is given by x 2 /(800)

2 + y

2 /(600)

2 =1, where x,

y in mm represent coordinates of any point on the track measured from point E. Neglect

friction. [Use formula for mentioned below]

B

40°

A, 6 kg

B, 10 kg

X

Y

E

B

500 mm 500 mm

P = 40 N

BONUS: Prove radius of curvature of an ellipse x 2 /a

2 + y

2 /b

2 =1 is given by the formula

| ( (

|

Further, show dy/dx = ±( ( √ ) (15)

7) The 0.6 kg collar slides on the curved rod in the vertical plane with negligible friction

under the action of a constant force F (as shown) in the cord guided by the small pulleys

at D. If the collar is released from rest at A, determine the force F which will result in the

collar striking the stop at B with a velocity of 4 m/s.

HINT: The non-conservative force F (i.e. the tension or the pull) will do work as the

collar goes from A to B. To compute this work, use the following steps

i) Choose a polar coordinate (er, eθ) system with its origin at D ii) Draw f.b.d of collar at any generic position between points A and B iii) Express F vectorially in that coordinate system

iv) Use the fact that ⃗ (Go back to your notes on polar coordinate system if you forgot what these terms/symbols mean)

v) Find ⃗ ⃗ and integrate over the path as collar goes from A to B to get the work done.

8) The 64.4 lb crate slides down a rough curved path in the vertical plane as shown. If the

crate has a speed of 3 ft/s at point A and a speed of 25 ft/s at B, compute the work done

by friction during the motion from A to B. [Hint: Work done by all non-conservative

forces acting on a particle = Change in KE + Change in PE]

9) A thin circular rod is supported in a vertical plane by a bracket at A. Attached to the

bracket and loosely wound around the rod is a spring of stiffness k = 40 N/m with

undeformed length equal to the arc of circle AB. A 200-g collar C, not attached to the spring,

can slide without friction along the rod. Knowing that the collar is released from rest when θ

= 30°, determine the maximum height above point B reached by the collar, (b) the maximum

velocity of the collar.

10) The spring constant k = 700 N/m. The masses mA = 14 kg and mB = 18 kg. The

horizontal bar is smooth. At the instant shown, the spring is unstretched and the mass B is

moving downward at 1 m/s. How fast is B moving when it has moved downward 0.2 m from

its present position? Neglect friction.