A child's pogo stick stores energy in a spring with a force constant of 2.5×104N/m. At position A, the spring
compression is a maximum and the child is momentarily at rest. At position B, the spring is in its equilibrium position 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 child and pogo stick is 25.0 kg. If xA is 10cm below the line shown find out the distance xC.
To find the distance xC, we need to use the concept of conservation of mechanical energy. At position A, the pogo stick stores potential energy in the spring. At position C, this potential energy is converted into kinetic energy. By equating these two energies, we can find xC.
The potential energy stored in the spring at position A can be calculated using the formula:
Potential energy = (1/2) * k * x^2
Where k is the force constant of the spring (2.5x10^4 N/m) and x is the compression of the spring. Given that xA is 10 cm below the line, xA = -0.1 m (negative sign indicates compression).
Potential energy at position A = (1/2) * (2.5x10^4 N/m) * (-0.1 m)^2
= 125 J
At position C, all the potential energy is converted into kinetic energy. The kinetic energy of an object can be calculated using the formula:
Kinetic energy = (1/2) * m * v^2
Where m is the mass of the object (25.0 kg) and v is the velocity. Since the child is momentarily at rest at position C, the velocity v is 0 m/s.
Kinetic energy at position C = (1/2) * (25.0 kg) * (0 m/s)^2
= 0 J
Since the mechanical energy is conserved, the potential energy at position A is equal to the kinetic energy at position C.
Potential energy at A = Kinetic Energy at C
125 J = 0 J
Since this equation is not satisfied, it means that the child cannot reach position C with a velocity of 0 m/s. Hence, it is not possible to determine the distance xC without additional information or assumptions about the motion of the child.