A child\'s pogo stick as shown below stores energy in a spring with a force cons

Question

A child's pogo stick as shown below stores energy in a spring with a force constant of 2.40 104 N/m. At position A (xA = -0.110 m), the spring compression is a maximum and the child is momentarily at rest. At position B (xB = 0), the spring is relaxed and the child is moving upward. At position C, the child is again momentarily at rest at the top of the jump. The combined mass of the child and the pogo stick is 25.0 kg. (a) Calculate the total energy of the child-stick-Earth system if both gravitational and elastic potential energies are zero for x = 0. J (b) Determine xC. m (c) Calculate the speed of the child at x = 0. m/s (d) Determine the value of x for which the kinetic energy of the system is a maximum. mm (e) Calculate the child's maximum upward speed. m/s

Explanation / Answer

The spring constant, k = 2.40 * 10^4 N/m The mass ( pogo stick + child ) = 26.0 kg The compression in the spring, x = 0.14 m
(a) At position A:
PE = -mg*(0.14)      = - 35.672 J
EE = 1/2 k * (0.14)^2      = 235.2 J
KE = 0

Total energy, T.E =199.528 J

(b) xC is when KE and EE = 0
i.e PE = mgxC = TE     xC = TE/mg          = 0.783 m

(c) When x = 0, EE = 0, and PE = 0
KE = TE = 1/2 mv^2
     v = (2*TE/m)^1/2            = 3.91 m/s (d)
To find maximum upward speed, we need to find maximum KE OR minimum PE+EE.
PE + EE = -mgx + 1/2 k * x^2 199.528 = -254.8x +12000 x^2    x = 0.14 m           = 14*10^5 mm (e)
Alternatively,
KE + PE + EE = constant
1/2 mv^2 + mgx + kx^2 /2 = constant
differentiate wrt time
mv * dv/dt - mg*v - kx*v = 0
When v is max, dv/dt = 0      

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