Directions: Assume, unless otherwise specified, that all numbers have at least 3

Question

Directions: Assume, unless otherwise specified, that all numbers have at least 3 significant figures. A spring that does not conform to Hooke's Law has a force whose magnitude is given by F = 4x + 2x^3N where x is the distance from the un-stretched length of the spring and the direction of the force is opposite to the displacement of the spring. One end of the spring is attached to a support and mass M_1 = 2 kg is attached to the other end. M_1 is also attached to M_2, a 4 kg mass, via a light string and a massless frictionless pulley. The surface upon which M_1 rests is frictionless. With the spring in the un-stretched position, M_2 is given a shove such that when M_2 has fallen a distance of 1.5 m its velocity is -3 m/s j. What is the speed of M_2 as it passes through the equilibrium point of the system? What is the maximum stretch of the spring? Repeat B if there is a frictional force between M_1 and the surface characterized by a coefficient of kinetic friction mu k= 0.4.

Explanation / Answer

work done by spring when displaced dx distance at x extension.

dW = -F.dx = 4x + 2x^3 dx

integrtaing,

W = - 2x^2 - 0.5x^4

now using Work energy theorem.

Work done by spring + work done by gravity = change in KE

-2(1.5^2) - 0.5(1.5^4) + (4 x 9.8 x 1.5) = (2+ 4)3^2 /2 + KEi

KEi = 24.77 J

(2 + 4) v^2 /2 = 24.77

v = 2.87 m/s .......Ans

B) maximum stretch will be when KE = 0

-2x^2 - 0.5x^4 + (4 *9.8*x) = 0 - 24.77

0.5x^4 + 2x^2 - 39.2x - 24.77= 0
x = 4.19 m

C)
now using Work energy theorem.

Work done by spring + work done by gravity + work done by friction = change in KE

-2(1.5^2) - 0.5(1.5^4) + (4 x 9.8 x 1.5) - (0.4 x 4 x 9.8 x 1.5)= (2+ 4)3^2 /2 + KEi

KEi = 1.25 J

maximum stretch will be when KE = 0

-2x^2 - 0.5x^4 + (4 *9.8*x) - (0.4*4*9.8*x) = 0 - 1.25

0.5x^4 + 2x^2 - 23.52x - 1.25 = 0

x = 3.26 m

Related Questions

Navigate

• Browse (All)
• Browse D
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.