Answer:
The force exerted by the spring on the block is given by Hooke's Law:
F = -kx
where F is the force, k is the spring constant, and x is the displacement from the equilibrium position. Since the spring is initially unstretched, x = 0 at t = 0. The negative sign indicates that the force is in the opposite direction to the displacement.
The initial velocity of the block is -10.0 m/s, so its initial kinetic energy is:
KE = (1/2) * m * v^2
KE = (1/2) * (2.00 kg) * (-10.0 m/s)^2
KE = 100 J
According to the law of conservation of energy, the total mechanical energy of the system (spring-block) is conserved. Therefore, at any time t, the sum of the kinetic and potential energies is constant:
KE + PE = constant
where PE is the potential energy stored in the spring, given by:
PE = (1/2) * k * x^2
where x is the displacement from the equilibrium position. Since the spring is initially unstretched, the potential energy is also zero at t = 0.
At any time t, the displacement x can be expressed in terms of the velocity v using the equation of motion:
v = dx/dt
where x is the displacement from the equilibrium position. Since the block is moving in the negative direction at t = 0, we have:
x = -vt
Substituting this expression for x into the potential energy equation, we get:
PE = (1/2) * k * (-vt)^2
PE = (1/2) * k * v^2 * t^2
Substituting the given values, we get:
PE = (1/2) * (400 N/m) * (10.0 m/s)^2 * t^2
PE = 2000 t^2 J
At any time t, the total mechanical energy is:
E = KE + PE
E = 100 J + 2000 t^2 J
The maximum potential energy is equal to the maximum kinetic energy, which occurs when the velocity is zero at the equilibrium position. Therefore, we can solve for the time at which the velocity is zero:
KE + PE = 100 J + 2000 t^2 J = 0
Solving for t, we get:
t = sqrt(-0.05) s
This result is not physically meaningful because it implies a negative time. This indicates that the block does not come to a stop at the equilibrium position but instead continues to oscillate back and forth with a periodic motion.
To find the amplitude of the motion, we can use the equation of motion for simple harmonic motion:
x = A * sin(ωt)
where A is the amplitude, ω is the angular frequency, and t is the time. The angular frequency is given by:
ω = sqrt(k/m)
Substituting the given values, we get:
ω = sqrt(400 N/m / 2.00 kg)
ω = 10 rad/s
To find the amplitude, we need to use the initial conditions. At t = 0, the block has a negative velocity and is moving in the negative direction. Therefore, the displacement at t = 0 is:
x = -A
Substituting this into the equation of motion, we get:
-A = A * sin(0)
-A = 0
Step-by-step explanation:
Therefore, the amplitude is zero. This means that the block does not oscillate but instead moves with a constant velocity of -10.0 m/s in the negative direction.