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Work can be calculated by $W = Fdcostheta$ where $theta$ is the angle between the force $F$ and the displacement $d$

Let's say there's a ball and a (physically ideal - no friction etc.) robotic arm situated in otherwise empty space. The arm takes the ball, moves it around in a circle, and then returns it exactly to where it started. The arm also returns to its starting position.

In this case, there is no displacement overall. The ball and the robot arm are in the exact same positions as when they began.

Thus, $W = Fcdot 0 cdot costheta = 0$ So no energy was required to move the ball in a circle.

However, this disagrees with my intuition, because if I were to make a robot arm that would do this, I feel like I would need to give it an energy source (for example, a battery), and that by the time the robot was done, I would have lost energy from that battery.

Does the movement require energy? If so, where did I mess up in my calculation? If not, why would my robot need a battery pack or some energy source in order to make this movement? (If it doesn't, please explain.)

First, you are not equating the work done correctly. This is a good physics lesson. Please understand your equations before you use them. Blindly plugging in numbers will not work out. The equation you give is only true for motion in one dimension and with a constant force. Plugging in $0$ for displacement is not correct here. In general you need to look at infinitesimal displacements $text dmathbf x$ and calculate the work $text d W=mathbf Fcdottext dmathbf x$, then integrate (add up) the total work.

Now, I am assuming the ball starts and stops at rest. Therefore, the arm does work to increase the ball's speed, and then it does the same amount of negative work to bring it to rest. So the net work is $0$, but it is because the total change in kinetic energy is $0$ (since $W=Delta K$), not because the displacement is $0$ around the circle.

Now, this is not the same thing as the robot using something like a battery. The robot (neglecting friction) has to apply forces to change the speed, and this requires power from the power supply. Just imagine yourself doing the action of the robot. You will need to exert effort to get the ball (and yourself) moving, and you will need to exert effort to get the ball (and yourself) to stop rotating.

The definition of work is energy = force * displacement * cos(theta)

This definition is only strictly accurate for movement on a straight line.
In general, you need to solve an integral

$$E = int_a^bvec{F}cdot dvec{ell}$$

Does the movement require energy?

Two reasons a real robot will require energy to do this operation:

• Real robots have friction in their joints, so there will always be a force opposing the direction the robot is trying to move, and so the integral of force along the path of motion won't go to zero




• If you are considering cases where the motion includes some components downward in relation to gravity and some components upward, not every robot will be designed to recover the energy gained by moving downward and use it to move the object back upward, so some energy will be used lifting the object on the upward part of the path.