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Chapter 3: Problem 31

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.

Short Answer

Expert verified
Question: Describe a situation in which the total angular momentum of a body is zero while it is both spinning and moving in a circle. Answer: A situation in which the body's total angular momentum is zero while it is spinning and moving in a circle can be achieved when its spin angular momentum and orbital angular momentum have equal magnitudes but opposite directions. This causes the spin angular momentum to completely cancel out its orbital angular momentum, resulting in zero total angular momentum.

Step by step solution

01

Understanding Angular Momentum

Angular momentum is a measure of the rotational motion of an object. It is a vector quantity and can be defined as the cross product of the radius vector (from the axis of rotation to the point on the body) and the linear momentum (mass multiplied by linear velocity): angular momentum (L) = r x m*v Where r is the radius, m is the mass, and v is the linear velocity of the object. Keep in mind, since it's a cross product, the direction of the angular momentum will be perpendicular to the plane of motion.
02

Two Types of Angular Momentum

In this problem, we are dealing with two types of angular momentum: (1) the angular momentum associated with the body's spinning motion around its own axis of rotation (spin angular momentum), and (2) the angular momentum associated with the body's motion in a circle around another fixed point (orbital angular momentum). Spin angular momentum (L_spin) and orbital angular momentum (L_orbit) can be calculated using the same equation L = r x m*v, but for each of them, we are considering different values of 'r' and 'v'.
03

Spin Angular Momentum

To calculate the spin angular momentum (L_spin) of the spinning body, we need to consider the radius vector from the body's axis of rotation to the point on the body (r_spin) and the tangential velocity of that point (v_spin). L_spin = r_spin x m*v_spin The direction of the spin angular momentum will be parallel to the body's axis of rotation.
04

Orbital Angular Momentum

To calculate the orbital angular momentum (L_orbit) of the body moving in a circle around a fixed point, we need to consider the radius vector from the center of the circle (fixed point) to the point on the body (r_orbit) and the linear velocity of that point along the circular path (v_orbit). L_orbit = r_orbit x m*v_orbit The direction of the orbital angular momentum will be perpendicular to the plane of the circular motion.
05

Condition for Total Angular Momentum to be Zero

The total angular momentum of the body (L_total) is the vector sum of its spin angular momentum (L_spin) and its orbital angular momentum (L_orbit): L_total = L_spin + L_orbit For the total angular momentum (L_total) to be zero, both magnitudes and directions of L_spin and L_orbit need to be equal in magnitude but opposite in direction. L_spin = -L_orbit This means that the body must be spinning in such a way that its spin angular momentum completely cancels out its orbital angular momentum.
06

Conclusion

In conclusion, it is possible for a body to be spinning and moving in a circle in such a way that its total angular momentum is zero. This can be achieved when the spin angular momentum (L_spin) and the orbital angular momentum (L_orbit) have equal magnitudes but opposite directions, thus canceling each other out.

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