Question

which most simplified form of the law of conservation of energy describes the motion of the block when it slides from the top of the table to the ramp? view available hint(s) (\frac{1}{2}mv_i^2 + mgh_i + w_{\text{nc}} = \frac{1}{2}mv_f^2 + mgh_f) (\frac{1}{2}mv_i^2 + \frac{1}{2}kx_i^2 = \frac{1}{2}mv_f^2 + \frac{1}{2}kx_f^2) (\frac{1}{2}mv_i^2 + mgh_i = mgh_f + \frac{1}{2}kx_f^2) (\frac{1}{2}mv_i^2 + mgh_i = \frac{1}{2}mv_f^2 + mgh_f) (\frac{1}{2}mv_i^2 + mgh_i + \frac{1}{2}kx_i^2 + w_{\text{nc}} = \frac{1}{2}mv_f^2 + mgh_f + \frac{1}{2}kx_f^2) submit

Explanation:

Step1: Recall Energy Conservation Basics

The law of conservation of energy (work - energy theorem) is \(\Delta K+\Delta U + W_{nc}=0\), or \(K_i + U_i+W_{nc}=K_f + U_f\), where \(K = \frac{1}{2}mv^{2}\) (kinetic energy), \(U\) is potential energy, and \(W_{nc}\) is work done by non - conservative forces. For gravitational potential energy, \(U = mgh\). If there is no spring (no elastic potential energy, \(\frac{1}{2}kx^{2}=0\)) and no non - conservative forces (like friction, so \(W_{nc} = 0\)) acting on the block as it slides from the top of the table to the ramp (assuming a smooth surface, so no friction), the energy conservation equation simplifies.

Step2: Analyze Each Option

• Option 1: \(\frac{1}{2}mv_{i}^{2}+mgh_{i}+W_{nc}=\frac{1}{2}mv_{f}^{2}+mgh_{f}\) includes \(W_{nc}\), but if there is no non - conservative force (e.g., frictionless motion), \(W_{nc} = 0\), so this is not the most simplified form.
• Option 2: \(\frac{1}{2}mv_{i}^{2}+\frac{1}{2}kx_{i}^{2}=\frac{1}{2}mv_{f}^{2}+\frac{1}{2}kx_{f}^{2}\) includes elastic potential energy (\(\frac{1}{2}kx^{2}\)), but there is no spring involved in the motion of the block sliding from the table to the ramp, so this is incorrect.
• Option 3: \(\frac{1}{2}mv_{i}^{2}+mgh_{i}=mgh_{f}+\frac{1}{2}kx_{f}^{2}\) includes elastic potential energy (\(\frac{1}{2}kx_{f}^{2}\)) which is not present, so incorrect.
• Option 4: \(\frac{1}{2}mv_{i}^{2}+mgh_{i}=\frac{1}{2}mv_{f}^{2}+mgh_{f}\) assumes \(W_{nc}=0\) (no non - conservative forces) and no elastic potential energy (no spring). This is the case for a block sliding (without friction, no spring) where kinetic energy (\(K=\frac{1}{2}mv^{2}\)) and gravitational potential energy (\(U = mgh\)) are conserved (since \(W_{nc} = 0\) and no elastic PE).
• Option 5: \(\frac{1}{2}mv_{i}^{2}+mgh_{i}+\frac{1}{2}kx_{i}^{2}+W_{nc}=\frac{1}{2}mv_{f}^{2}+mgh_{f}+\frac{1}{2}kx_{f}^{2}\) includes elastic potential energy and non - conservative work, which are not present in the simple sliding motion from table to ramp.

Answer:

\(\boldsymbol{\frac{1}{2}mv_{i}^{2}+mgh_{i}=\frac{1}{2}mv_{f}^{2}+mgh_{f}}\) (the fourth option in the list)