- #1

- 665

- 26

## Main Question or Discussion Point

Torque ##\tau## should be in the same direction as the change in angular momentum ##\Delta L##, but the following example seems to suggest otherwise.

Consider a cone rolling on its side without slipping on a flat surface. Let the apex be the origin and the initial coordinate of the center of mass (CM) of the cone be (-a, 0, c), where a and c are positive constants. (That means the cone, or rather its projection, is in the middle of the 2nd and 3rd quadrant of the x-y plane.) The cone rotates anticlockwise about the z axis. So, when viewed from the apex to the base, the cone rotates anticlockwise around its symmetry axis. That means the angular velocity ##\omega## and hence the angular momentum ##L## points in the +x direction, if we ignore their z components.

At this instant, the CM of the cone will move in the -y direction, making ##L## point into the first quadrant in the next instant. Thus, ##\Delta L## is in the +y direction.

Since the CM moves in a circle, there must be a friction ##F## to provide the centripetal force. Thus, at this instant, the friction acts in the +x direction. The fiction acts along the line of contact, which is vertically below the CM. So the torque ##\tau=r\times F## about the CM is in the -y direction, which is opposite to ##\Delta L##.

What's wrong?

Consider a cone rolling on its side without slipping on a flat surface. Let the apex be the origin and the initial coordinate of the center of mass (CM) of the cone be (-a, 0, c), where a and c are positive constants. (That means the cone, or rather its projection, is in the middle of the 2nd and 3rd quadrant of the x-y plane.) The cone rotates anticlockwise about the z axis. So, when viewed from the apex to the base, the cone rotates anticlockwise around its symmetry axis. That means the angular velocity ##\omega## and hence the angular momentum ##L## points in the +x direction, if we ignore their z components.

At this instant, the CM of the cone will move in the -y direction, making ##L## point into the first quadrant in the next instant. Thus, ##\Delta L## is in the +y direction.

Since the CM moves in a circle, there must be a friction ##F## to provide the centripetal force. Thus, at this instant, the friction acts in the +x direction. The fiction acts along the line of contact, which is vertically below the CM. So the torque ##\tau=r\times F## about the CM is in the -y direction, which is opposite to ##\Delta L##.

What's wrong?