I completely understand the concepts behind uniform circular motion. But let's say you are spinning a ball connected by a string to a motor in a horizontal circle. When increasing the angular velocity of the spinning motor, I can't see how the ball connected to the string will have any force that allows it to increase its tangential velocity. How would the string be able to pull it so it accelerates tangentially all while undergoing circular motion? An example would be if you are swinging a ball in a circle above your head and you begin to spin it faster. How is the string causing the object to increase its linear speed? I believe tension can't cause it unless it's working at an angle less than 90 to the tangent because work must be done to increase the kinetic energy.
Circular Motion – How Does an Object Undergoing Circular Motion Increase Its Tangential Velocity?
centripetal-forceclassical-mechanicsforcesnewtonian-mechanics
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Your equation is wrong because $V_{tangent}$ varies along the rod. It should be inside the integral.
Finding the tension in the string by the 2nd method is much more sensible. All you need to do is apply the usual formula for centripetal force : $T=Mv^2/R$ where $R$ is the radius of the circle traced out by the CM of the rod.
It is Newton's first law:
In an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.
If by some magic, the attraction of the sun stopped, the earth would leave off on a tangent.
It all started with the formation of the solar system:
Scientists believe that the solar system was formed when a cloud of gas and dust in space was disturbed, maybe by the explosion of a nearby star (called a supernova). This explosion made waves in space which squeezed the cloud of gas and dust. Squeezing made the cloud start to collapse, as gravity pulled the gas and dust together, forming a solar nebula. Just like a dancer that spins faster as she pulls in her arms, the cloud began to spin as it collapsed. Eventually, the cloud grew hotter and denser in the center, with a disk of gas and dust surrounding it that was hot in the center but cool at the edges. As the disk got thinner and thinner, particles began to stick together and form clumps. Some clumps got bigger, as particles and small clumps stuck to them, eventually forming planets or moons . Near the center of the cloud, where planets like Earth formed, only rocky material could stand the great heat
The initial forces were probably thermodynamic exchanges of scatterings . The slow domination of the collective gravitational field condensing to a sun and planets again depends on laws : conservation of angular momentum in particular.
So as implied in the comments, the real question is "why Newton's first law".
And the answer is that laws and postulates in physics are the extra axioms imposed so that the mathematical theory fits the data. It explains the tangential velocity, but the law itself was chosen so that the kinematics would be fitted and new set ups could be explained and predicted. That is what physics is about, understanding mathematically the way nature works.
Best Answer
In the case that you describe, an individual swinging a mass horizontally on the end of a string, the string does not run directly to the centre of rotation. Instead, it runs to your hand, which in turn is moving in a circle about its centre of rotation . Sometimes the arm is involved, sometimes only a rotation at the wrist. ( Mime winding up a sling to throwing speed to see what I mean)
If everything is constant (and there's no drag on the mass), the line from mass to hand to centre of rotation is straight; the string tension exerts only the centripetal force needed to maintain the circle, as well as an upward component to keep the mass from dropping downward.
If you then speed up the circular motion of your hand to a new constant angular velocity, your hand's angular motion gets ahead of the mass's angular motion, and gets continuously farther ahead. So now the tension in the string is not in the line from the mass to the centre of rotation. There is a tangential component to the tension, constantly increasing, which serves to speed up the rotation of the mass.
This tangential force speeds up the mass's rotation, until the mass is rotating faster than your hand. Then the mass gets ahead, and your hand slows it down, and so on.
Picture a pendulum clock in a wheel-style space station, with the pendulum set to swing in the plane of the wheel, but currently at rest. Then speed up the space station slightly...