Question

Consider the block–spring–surface system in part (b) of example 8.6. Using an energy approach, find the position of the block at which its speed is a maximum.

Options:

At the equilibrium position
At the maximum compression of the spring
At the maximum extension of the spring
At the halfway point between equilibrium and maximum compression

Answer 1

Using an energy approach in a frictionless system where mechanical energy is conserved, the speed of the block is a maximum at the equilibrium position where the potential energy is zero and all of it has been converted to kinetic energy.

Step-by-step explanation:

When considering a block attached to a spring on a frictionless surface, the position at which the block's speed is a maximum can be determined using an energy approach. As the block moves through the equilibrium position (where the spring is neither compressed nor extended), all the potential energy of the spring has been converted to kinetic energy. According to the conservation of mechanical energy, the kinetic energy is maximum at this point because it contains all the energy that was initially potential energy in the spring when it was compressed or extended.

The relevant principle here is that in a frictionless system, mechanical energy is conserved, and this means that the total energy (potential + kinetic) remains the same throughout the motion. As such, when the potential energy is zero at the equilibrium position, the kinetic energy must be at its maximum value.

Therefore, the speed of the block is a maximum at the equilibrium position, where the spring force is zero and it is neither compressed nor stretched.