A toy race car travels around a vertical circle on a curved length of track without ever losing contact?

Question

Where is the car when its radial acceleration is smallest?explain..
where is the car when its tangential accelerationis smallest?Explain.
When the car is at the top of the circle, is the acceleration larger than, smaller than, or the same as the acceleration of an object falling freely in a gravitational field?Explain.

Answer 1

The car has its smallest radial acceleration at the bottom of the circle, because the centripetal acceleration is upwards while the acceleration due to gravity is downwards; since they act in opposite directions, their magnitudes are subtractive, and total magnitude is smaller.

The tangential acceleration is greatest when the car is moving directly dowards, which occurs at the 270 degree position (if the bottom is 0 degrees and it moves through 90 degrees and then to the top at 180 degrees). This is because the centripetal acceleration has no tangential component, so the acceleration due to gravity provides the only tangential component. At 90 and 270 degrees, the downwards force of gravity is actually tangential to the direction of motion, maximizing the tangential acceleration. At 90 degrees, the car is moving upwards, so the force of gravity causes a deceleration. But at 270 degrees, the car is moving downwards and gravity causes it to speed up at its maximum rate.

At the top of the circle, the acceleration is larger than the acceleration of an object falling freely in a gravitational field, because the acceleration is equal to g + (v^2 / r), where g is the free-fall acceleration and v^2 / r is the centripetal acceleration, which will also be downwards. So as long as v is greater than zero, which it must be in order to continue moving around the circle, the acceleration will be larger than that of free-fall.