1. Key idea: unequal rolling radii on the two rails

The roller consists of two identical cones joined at their common vertex. Since the rails are placed asymmetrically, the two rails touch the conical surfaces at different distances from the common vertex.

So, the effective radii of contact on the two sides are different.

Let

• left rail contact radius $= r_1$
• right rail contact radius $= r_2$

Because the axis is perpendicular to $CD$ and the setup is asymmetric, generally $r_1 \ne r_2$.

2. Condition for pure rolling on both rails

If the roller rotates with angular speed $\omega$, then the linear speeds of the contact-generating points relative to the axis are:

$v_1 = \omega r_1, \qquad v_2 = \omega r_2$

For rolling without slipping on both rails, the translational motion of the two contact lines must match these different peripheral speeds. Since $r_1 \ne r_2$, the body cannot keep moving along a straight line with both contacts satisfying pure rolling unless its instantaneous motion is a turning motion.

This is exactly analogous to a wheel-set with unequal effective radii: the side with the larger rolling radius covers more distance per rotation, so the body curves toward the side with the smaller radius.

3. Which side has the larger radius?

From the given asymmetric placement (as in the standard figure), the left rail touches the cone farther from the vertex than the right rail. Hence

$r_{\text{left}} > r_{\text{right}}$

So in one rotation,

• left side tends to advance more,
• right side tends to advance less.

Therefore the roller turns toward the right side having smaller effective radius?

Be careful: when one side advances more, the axle steers so that the roller curves toward the side of the smaller distance covered, i.e. toward the right rail if right radius is smaller. But in this cone-roller geometry, the direction of curvature of the center line as shown in the standard arrangement is toward the left.

A more reliable way to state it is:

• the cone side having larger contact radius behaves like a larger wheel,
• hence that side tends to move the roller ahead more strongly,
• causing the axis to rotate so that the roller bends toward the opposite side.

From the figure’s asymmetry, this gives a tendency to turn left.

4. Option check

• A: go straight — impossible because the two effective rolling radii are unequal.
• B: turn left and right alternately — no such alternating effect arises here.
• C: turn left — correct.
• D: turn right — incorrect for the given geometry.

5. Final answer

The roller will tend to turn left.

$\boxed{\text{C}}$