It is possible for a body to be both spinning and moving in a circle in such a way that its total angular momentum is zero. Describe how this can be.

Rotational angular momentum is the angular momentum of the spinning body itself. It can be calculated by multiplying the rotational inertia (I) of the body by its angular velocity (ω) in a vector form: L_rot = Iω. Orbital angular momentum, on the other hand, is the angular momentum of the body moving in a circle. It can be calculated by multiplying the linear momentum (m * v, where m is the mass and v is the velocity) of the body by the distance (r) to the center of the circle: L_orb = m * v * r. Keep in mind that these are vector quantities as well.

In order for the total angular momentum to be zero, the rotational and orbital angular momentum must have equal magnitudes and opposite directions: |L_rot| = |L_orb|. This means that, if one is clockwise, the other must be counterclockwise.

Consider an object of mass m and radius R, spinning at an angular velocity ω and revolving at a linear velocity v around the circle with radius r. Let's assume that both the spinning and revolving are taking place in the same plane for simplicity. To achieve a total angular momentum of zero, the object must be spinning and moving in a circle in such a way that: L_rot = -L_orb Iω = -(m * v * r) Now, let's assume that: - The object is spinning clockwise. - Its moment of inertia (I) is constant. - The object is moving counterclockwise along the circular path. In this situation, if the magnitudes of the rotational and orbital angular momentum are equal, the total angular momentum will be zero. To illustrate this, let's assume an object is spinning at a rate of 5 rad/s clockwise, with a moment of inertia (I) of 2 kg * m^2. Its rotational angular momentum will be: L_rot = Iω = 2 kg * m^2 * 5 rad/s = 10 kg * m^2 * rad/s (clockwise) Now, the object is orbiting a circular path with a radius (r) of 1 m. To cancel out the clockwise rotational angular momentum, we need an equal counterclockwise orbital angular momentum: L_orb = m * v * r = -L_rot => v = 10 kg * m^2 * rad/s / (m * r) Assuming a mass (m) of 1 kg, we find that the object must move counterclockwise around the circle with a linear velocity (v) of 10 m/s: v = 10 kg * m^2 * rad/s / (1 kg * 1 m) = 10 m/s (counterclockwise) In this case, the object is both spinning clockwise and moving counterclockwise around a circle, resulting in a total angular momentum of zero, as demonstrated by the equal magnitudes and opposite directions of L_rot and L_orb.