A disc of radius R and mass M is rolling horizontally without slipping with speed . It then moves up an inclined smooth surface as shown in figure. The maximum height that the disc can go up the incline is :
The object is rolling with velocity \( V_0 \) and angular velocity \( \omega \). The reference line is taken such that the potential energy \( U = 0 \) at the surface of point B/C, which is smooth. Also, the relationship between velocity and angular velocity for rolling motion is given by:
\[
V_0 = \omega R
\]
Step 1: Conservation of Energy:
The principle of conservation of energy is applied:
\[
K_i + U_i = K_f + U_f
\]
where:
- \( K_i \) and \( U_i \) are the initial kinetic and potential energies,
- \( K_f \) and \( U_f \) are the final kinetic and potential energies.
Step 2: Applying the energy equation:
The initial and final energies are expressed as:
\[
\frac{1}{2}mv_0^2 + \frac{1}{2} \left( \frac{MR^2}{2} \right) \left( \frac{v_0}{R} \right)^2 + 0 = mgh + \frac{1}{2} \left( \frac{MR^2}{2} \right) \left( \frac{v_0^2}{R^2} \right)
\]
Step 3: Simplifying the equation:
Simplifying the above equation:
\[
\frac{1}{2} mv_0^2 = mgh_{\text{max}}
\]
Step 4: Finding the maximum height:
Solving for \( h_{\text{max}} \):
\[
h_{\text{max}} = \frac{v_0^2}{2g}
\]
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Approach Solution -2
Given: - Mass of the disc: \( M \) - Radius of the disc: \( R \) - Speed of the disc: \( v \) - Gravitational acceleration: \( g \)
Step 1: Total Kinetic Energy of the Rolling Disc
For a disc rolling without slipping, the total kinetic energy (\( K \)) is the sum of translational kinetic energy and rotational kinetic energy:
\[ K = \text{Translational K.E.} + \text{Rotational K.E.} \] \[ K = \frac{1}{2} M v^2 + \frac{1}{2} I \omega^2 \]
The moment of inertia (\( I \)) of the disc about its center is:
\[ I = \frac{1}{2} M R^2 \]
The angular velocity (\( \omega \)) is related to the linear speed by:
\[ \omega = \frac{v}{R} \]
Substituting these values:
\[ K = \frac{1}{2} M v^2 + \frac{1}{2} \left(\frac{1}{2} M R^2\right) \left(\frac{v}{R}\right)^2 \] \[ K = \frac{1}{2} M v^2 + \frac{1}{4} M v^2 \] \[ K = \frac{3}{4} M v^2 \]
Step 2: Conservation of Energy
As the disc moves up the incline, its kinetic energy is converted into potential energy (\( U \)) at the maximum height \( h \):
\[ K = U \] \[ \frac{3}{4} M v^2 = M g h \]
Solving for \( h \):
\[ h = \frac{3}{4} \frac{v^2}{g} \]
Conclusion:
The maximum height that the disc can go up the incline is \( \frac{3}{4} \frac{v^2}{g} \).
