a) When the acceleration of a mass on a spring is zero, the velocity is at a maximum
b) When the velocity of a mass on a spring is zero, the acceleration is at a maximum
Explanation:
a)
In a mass-spring system, there is continuous conversion of energy from kinetic energy (K) to elastic potential energy (U). The total mechanical energy at any moment of the motion is given by
[tex]E=K+U=\frac{1}{2}mv^2+\frac{1}{2}kx^2[/tex]
where
m is the mass
v is the velocity
k is the spring constant
x is the displacement
Since the total energy E must remain constant, it means that when the displacement (x) is minimum, the velocity (v) is maximum, and viceversa. Moreover, the acceleration in a mass-spring system is related to the displacement by the equation
[tex]a=-\omega^2 x[/tex]
where [tex]\omega[/tex] is the angular frequency of the system. From this equation, we see that the acceleration is directly proportional (in its absolute value) to the displacement: so, when the acceleration of the mass is zero, the displacement is also zero, and this means that the velocity of the mass is at its maximum. This occurs when x = 0, so when the mass passes through the equilibrium position of the spring.
b)
We can use a similar argument to answer part b) of this question. In fact, we said that:
- When the velocity is zero, the displacement is at its maximum
- The acceleration is directly proportional to the displacement (in its absolute value), according to [tex]a=-\omega^2 x[/tex]
Therefore, we can say that when the velocity is zero, the acceleration of the mass-spring system is at maximum. This occurs when the displacement of the system is maximum, so when the spring is totally stretched/compressed, and the velocity of the system at that moment is zero.
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