three carts of masses 2 kg, 18 kg, and 9 kg move on a frictionless horizontal track with speeds of 10m/s, 8m/s, and 2m/s. the ca
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1) 256.9 N/m
The force applied to the spring is equal to the weight of the block hanging on the spring:
And the spring constant can be found by using Hook's law, because we know that the displacement caused by this force is x = 0.29 m:
2) 1.08 Hz
The angular frequency of oscillation of the spring is given by the formula:
And the frequency of oscillation is given by:
3) 2.19 m/s
The velocity at time t of the block is given by:
where
is the initial velocity of the block
is the angular frequency
t is the time
Substituting t=0.36 s, we find the speed of the block at that time:
And the negative sign means that the direction of the velocity is upward (because the initial velocity was downward)
4) 25.6 m/s^2
The maximum acceleration is given by:
where A is the amplitude of the oscillation.
We can find the amplitude by using the law of conservation of energy: in fact, the kinetic energy at the equilibrium point must be equal to the elastic potential energy at the point of maximum displacement:
So, the maximum acceleration is
5) 95.1 N
The magnitude of the net force acting on the block is given by the difference between the weight and the restoring force of the spring:
First, we need to find the position x at t=0.36 s, which is given by
And so, the net force is
And the negative sign means the direction of the force is upward.