In Fig. 8-22, a block slides from A to C along a frictionless ramp, and then it passes through horizontal region CD, where a frictional force act on it. Is the block’s kinetic energy increasing, decreasing, or constant in (a) region AB, (b) region BC, and (c) region CD? (d) Is the block’s mechanical energy increasing, decreasing, or constant in those regions?

A figure which shows the block slides from A to C along a frictionless ramp and then it passes through the horizontal region CD

The problem is based on the principle of conservation of energy, which states that the total energy in an isolated system remains constant. Here this principle can be used to find whether the block’s kinetic energy increases, decrease, or remains constant in the region$\mathbf{AB}\mathbf{,}\mathbf{}\mathbf{BC}\mathbf{}\mathbf{}\mathbf{and}\mathbf{}\mathbf{CD}$. Also, it can be used to find whether the mechanical energy of the block is increasing, decreasing, or remains constant in$\mathbf{AB}\mathbf{,}\mathbf{}\mathbf{BC}\mathbf{}\mathbf{}\mathbf{and}\mathbf{}\mathbf{CD}$region.

Formula:

role="math" localid="1657184859975" ${\mathrm{\Delta E}}_{\mathrm{mec}}=\mathrm{d}(\mathrm{K}.\mathrm{E})+\mathrm{dU}$

For the conservation of energy,

$$\begin{gathered} {\mathrm{\Delta E}}_{\mathrm{mec}}=\mathrm{d}(\mathrm{K}.\mathrm{E})+\mathrm{dU} \\ =\mathrm{constant} \end{gathered}$$

As the height of the ramp decreases from A to B, the potential energy of the block decreases. According to the above equation, as potential energy decreases, kinetic energy increases.

Therefore, the block’s kinetic energy increases in the region AB

For conservation of energy

$$\begin{gathered} {\mathrm{\Delta E}}_{\mathrm{mec}}=\mathrm{d}(\mathrm{K}.\mathrm{E})+\mathrm{dU} \\ =0 \end{gathered}$$

Velocity of the block decreases when it comes to point B, therefore its kinetic energy decreases. As the kinetic energy decreases, the potential energy increases.

Hence, the block’s kinetic energy decreases in the region BC.

As region CD is plane and has friction, the velocity of the block decreases when it slides through this region. Therefore, its kinetic energy decreases in the region CD

According to this principle, when kinetic energy increases in the region AB and BC, its potential energy decreases. But, in the region CD due to friction, the kinetic energy of the block increases, and because it is a plane, the potential energy of the block remains constant.

Therefore, the mechanical energy of the block remains constant in the region AB & BC but decreases in the CD region