The acceleration is at its maximum value when A. when the mass has a speed of zero
Explanation:
We can answer this question by using the law of conservation of energy: in fact, the total mechanical energy of the spring (elastic potential energy + kinetic energy) is constant during the motion. So we can write:
[tex]E=K+U=\frac{1}{2}mv^2+\frac{1}{2}kx^2 =constant[/tex] (1)
where
m is the mass
v is the velocity
k is the spring constant
x is the displacement of the mass from the equilibrium position of the spring
We also note that the restoring force in the spring is (its absolute value)
[tex]F=kx[/tex]
Also, according to Newton's second law, the force can be written as product of mass (m) and acceleration (a), so we get:
[tex]ma=kx\\x=\frac{m}{k}a[/tex]
Therefore we can rewrite eq.(1) as
[tex]E=K+U=\frac{1}{2}mv^2+\frac{1}{2}k(\frac{m}{k})^2a^2 =constant[/tex]
From the equation, since E must remain constant, we notice that:
- When the speed v is the maximum, the acceleration a is minimum (zero)
- When the speed v is zero (this occurs when the spring is at its maximum displacement), the acceleration a is maximum
Therefore, the acceleration is at its maximum value when the mass has a speed of zero.
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