Question

A projectile of mass m is moving in the vertical x-y plane with the origin on the ground and y-axis pointing vertically up. Taking the gravitational potential energy to be zero on the ground, the total energy of the particle written in planar polar coordinates (r, θ) is (here g is the acceleration due to gravity)

• A. \(\frac{m}{2}ṙ^2+mgr\sin\theta\)
• B. \(\frac{m}{2}(ṙ^2+r^2\dot\theta^2)+mgr\cos\theta\)
• C. \(\frac{m}{2}(ṙ^2+r^2\dot\theta^2)+mgr\sin\theta\)
• D. \(\frac{m}{2}(ṙ^2+r^2\dot\theta^2)-mgr\cos\theta\)

Answer 1

The Correct Option is C

To determine the total energy of a particle in planar polar coordinates, we need to consider both the kinetic and potential energy of the particle.

The kinetic energy (T) of a particle in polar coordinates is given by:

\( T = \frac{1}{2} m (\dot{r}^2 + r^2 \dot{\theta}^2) \)

where \(\dot{r}\) is the radial velocity and \(r \dot{\theta}\) is the tangential velocity.

The potential energy (V) of the particle due to gravity, taking the zero level at the ground, is given by:

\( V = mgr \sin \theta \)

where \(\theta\) is the angle made with the vertical, and thus \(\sin \theta\) represents the height component in terms of r.

Therefore, the total energy (E) of the particle, which is the sum of the kinetic and potential energies, is:

\( E = T + V = \frac{1}{2} m (\dot{r}^2 + r^2 \dot{\theta}^2) + mgr \sin \theta \)

This matches with the correct option:

\(\frac{m}{2}(ṙ^2+r^2\dot\theta^2)+mgr\sin\theta\)

Let's analyze why the other options are incorrect:

• \(\frac{m}{2}ṙ^2+mgr\sin\theta\): This option omits the tangential kinetic energy term \(r^2\dot\theta^2\), making it incomplete.
• \(\frac{m}{2}(ṙ^2+r^2\dot\theta^2)+mgr\cos\theta\): This option incorrectly uses \(\cos \theta\) instead of \(\sin \theta\) for potential energy.
• \(\frac{m}{2}(ṙ^2+r^2\dot\theta^2)-mgr\cos\theta\): This option incorrectly uses \(\cos \theta\) and also incorrectly changes the sign of the potential energy term.

Thus, the correct answer is the option that includes both kinetic energy terms and the correct form of potential energy.