A block–spring system vibrating on a frictionless, horizontal surface with an amplitude of 2.0 cm has an energy of 14 J. If the block is replaced by one whose mass is twice the mass of the original block and the amplitude of the motion is again 2.0 cm, what is the energy of the system?
Question
A block–spring system vibrating on a frictionless, horizontal surface with an amplitude of 2.0 cm has an energy of 14 J. If the block is replaced by one whose mass is twice the mass of the original block and the amplitude of the motion is again 2.0 cm, what is the energy of the system?
Solution
The energy of a vibrating block-spring system is given by the formula:
E = 1/2 * m * w^2 * A^2
where: E is the energy, m is the mass of the block, w is the angular frequency, A is the amplitude of the motion.
Given that the energy E is 14 J when the mass m is m1 and the amplitude A is 2.0 cm, we can write:
14 J = 1/2 * m1 * w^2 * (2.0 cm)^2
If the block is replaced by one whose mass is twice the mass of the original block (m2 = 2*m1) and the amplitude of the motion is again 2.0 cm, the energy E2 of the system is:
E2 = 1/2 * m2 * w^2 * (2.0 cm)^2 = 1/2 * 2*m1 * w^2 * (2.0 cm)^2 = 2 * (1/2 * m1 * w^2 * (2.0 cm)^2) = 2 * 14 J = 28 J
So, the energy of the system when the block is replaced by one whose mass is twice the mass of the original block and the amplitude of the motion is again 2
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