A 2.0 kg mass is placed in front of a spring (K = 40 N/m) which has been compressed to its maximum of 20.0 cm.
(a)If the spring is released, determine the velocity of the mass at the instant that the spring is only compressed 10.0 cm.
(b) Determine the velocity of the mass the instant it leaves the spring.
To determine the velocity of the mass in both scenarios, we can use the principle of conservation of mechanical energy. The equation for potential energy stored in a spring is given by:
PE = 0.5 * K * delta_x^2
where PE is the potential energy, K is the spring constant, and delta_x is the displacement of the spring from its equilibrium position.
(a) The initial potential energy of the spring when it is compressed by 20.0 cm can be computed as:
PE_initial = 0.5 * 40 N/m * (0.2 m)^2
To find the velocity when the spring is compressed by 10.0 cm, we can equate the initial potential energy to the final kinetic energy, since energy is conserved:
PE_initial = KE_final
0.5 * 40 N/m * (0.2 m)^2 = 0.5 * m * v^2
Simplifying the equation and solving for velocity:
v = sqrt((2 * PE_initial) / m)
Substituting the values:
v = sqrt((2 * 0.5 * 40 N/m * (0.2 m)^2) / 2 kg)
(b) To determine the velocity of the mass the instant it leaves the spring, we consider conservation of mechanical energy again. When the mass leaves the spring, all the potential energy is converted into kinetic energy:
PE_final = 0
KE_initial = KE_final
0.5 * K * delta_x^2 = 0.5 * m * v^2
Substituting the values and solving for velocity:
v = sqrt((2 * 0) / m)
v = 0 m/s
Therefore, the velocity of the mass at the instant it leaves the spring is 0 m/s.
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