A chld\'s pogo stick (figure below) stores energy in a spring (k-3.35 x 104 N/m)

Question

A chld's pogo stick (figure below) stores energy in a spring (k-3.35 x 104 N/m). At posiion( -0.100 m), the spring compression is a maximum and the child is momentanily at rest. A osn(x,ting is relaxed and the child is moving upward. At position © the child is again momentarily at rest at the top of the ump. Assume that the combined mass of Child and pogo stick is 290g. (a) Calculate the total energy of the system if both potential energles are zero atx0. (b) Determine x2 0 49 1T (c) Calculate the speed of the child at x = 0. 3 10m/s (d) Determine the value of x for which the kinetic energy of the system is a maximum. (e) Obtain the child's maximum upward speed.

Explanation / Answer

(A) spring KE = k x^2 / 2

= (3.35 x 10^4)(0.10^2)/2

= 167.5 J  

gravitational PE = m g h

= (29 x 9.8 x - 0.10)

= - 28.42 J

totaL PE = 167.5 - 28.42 = 139 J

(b) 139 = m g h

h = 139 / (29 x 9.8) = 0.49 m

(C) at x = 0 , PE = 0

so KE = 139 J  

29 v^2 /2 = 139

v = 3.10 m/s

(D) for x, (below the x = 0 or for negative x)

PE = k x^2 /2 + m g x
  

KE = 167.5 - (k x^2 /2 + m g x )

  

differentiate wrt x ,

d(KE)/dx = m g + k x = 0

x= -m g / k = -8.5 x 10^-3 m

so KE will be max at

x = - 8.5 x 10^-3 mm Or - 8.5 mm

Ans: -8.5  

(E) KE = 167.5 - (29 x 9.8 x -8.5 x 10^-3) - (3.35 x 10^4 (-8.5 x 10^-3)^2)/2

KE = 168.7

29 v^2 /2 = 168.7

v = 3.41 m/s

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