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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.

If we can neglect external gravitational or electromagnetic fields then we only need to consider the kinetic energy of the ball. This is zero at the beginning of the motion and zero at the end, so the net change of energy in the ball is zero.

If we assume an ideal robot arm (no friction, perfect conductors, no air resistance etc.) then the energy that the robot arm puts into the ball to accelerate it at the start of the motion can be 100% recovered when the ball decelerates at the end of motion. So the net loss of energy from the robot arm is also zero.

In practice the robot arm will lose energy due to friction, resistive heating, air resistance etc.

If the robot arm is extended holding the ball and rotated horizontally, then the energy required would be calculated using torque, using the distance of the centre of mass and the total mass being moved by the motor.

Let us say that with the arm extended the centre of mass is 0.5m (distance from axis) and the total mass of arm + ball = 5kg, the torque rating is 10 Nm. Because torque is a rotational equivalent of linear motion then power = torque x 2 x pi x rotational speed (revs per second) (https://en.wikipedia.org/wiki/Torque)

Therefore, if we say the arm takes 1 second to complete the circle, the power required is 10 x 2 x pi x 1 = 62.83 Watts (= joules/sec). A typical stepper motor for a robot arm with a reach about 1m would draw 12V, therefore would have to be rated at 6A which in a perfect system would be capable of delivering 72 Watts.

NO, an ideal robot can move the load in a circle (actually in an any closed curve, not just a circle), producing zero total job and hence without any loss of the energy. However in order to do that, the robot must:

• Recover back all potential (lifting up the load) and kinetic (accelerating the load from zero initial velocity so it can be moved) energy without the loss. A simple electric engine can do this, but obviously not without the loss.


• Move the load without friction (well, you have stated in the question your robot can do this but even spacecraft do hit one another atom on they way through the vacuum).

The robot will need the energy in general, especially if the circle stands vertically like a Ferris wheel. Just an ideal robot will be able to recover all the energy back at the end of the loop.

A great example of such a machine is the Buzz Aldrin cycler. It is a "space bus" that travels between Earth and Mars essentially for free because it travels in a closed loop. In space, it is relatively easy to satisfy the two conditions above in a significant degree.

The infinitesimal work on an object is $Fdcosθ$. If any of those quantities aren't constant, then we have to take the integral over some path. However, all paths will the same starting and ending conditions will yield the same answer. If the ball ends up in the same condition that its started in, then it hasn't had any work done on it. So this is not the central issue.

What is the central issue is that this is the formula for doing work on an object. Just because the robot arm has done zero work on the ball, that does not mean that the robot arm has expended no energy. It just means that none of the energy expended by the robot arm has gone towards permanently increasing the kinetic energy of the ball. The robot arm could have expended energy otherwise, such as overcoming friction in its internal mechanisms. If the robot arm accelerated to move the ball, and then decelerated to stop the ball, then it took energy to accelerate, and then the energy went somewhere when the arm decelerated. If the deceleration happened through friction, then the energy dissipated into heat. But the robot arm could have regenerative braking, in which case some of the energy went back into the battery.

In any real world system, there will be some loss of energy to heat. No motor and no regenerative braking system operates with perfect efficiency. But in an ideal system with no friction or other inefficiency, the robot arm could indeed move the ball and end up with the same amount of energy in its battery, and thus this would not use up any energy. However, we would still have to have some energy to start with to power the system, even if that energy isn't used up.

Interesting question and a simple answer here. Newtonian first law claims, that:
$$frac{text d mathbf v}{text d t} = 0$$ In words it would be:
If net force applied to an object is zero, then object's speed must be constant

Your object moves in a circle and that means, that it changes speed, because speed is a vector which has magnitude and direction:

So if direction of movement changes, speed changes also. And if speed changes - there MUST be a force acting upon an object. And for keeping force, you need an energy of course.