The ball has a weight of 15 lb and is fixed to a rod having a negligible mass. If it is released

The ball has a weight of 15 lb and is fixed to a rod having...

Key Concepts

Critical Condition for Zero Reaction Force

The problem hinges on finding the specific condition when the compressive (reaction) force from the rod is exactly zero. At this juncture, the entirety of the centripetal force needed is provided solely by the radial component of the gravitational force. Establishing this critical condition is crucial as it marks the transition point where the rod no longer influences the centripetal acceleration of the ball.

Force Decomposition in Circular Motion

Analyzing the motion of the ball involves breaking down the forces into radial and tangential components. The weight of the ball can be decomposed such that its radial component contributes to the centripetal acceleration, whereas the tangential component affects the speed. Understanding this decomposition is essential to determine the conditions under which the rod’s compressive force becomes zero.

Conservation of Energy

This concept involves equating the gravitational potential energy lost to the kinetic energy gained by the ball as it moves along its circular path. By applying conservation of energy, one can relate the initial release height (or angle) to the speed of the ball at any subsequent position, which is critical when examining the forces acting on the ball during its motion.

Centripetal Force Requirement

In circular motion, the object must experience a net inward force (centripetal force) to keep it moving along a curved path. This force depends on the square of the velocity and the radius of the circle. Determining the point where the rod no longer exerts a compressive force requires equating the gravitational component acting radially inward to the centripetal force needed to sustain the motion.

Transcript

00:01 So on this problem, we have a ball connected to some rod, and we want to find the angle when the compressive force on the ball is zero.

00:09 So we'll say we'll call that f, and we're given the weight of the ball, it's 15 pounds.

00:19 Let's convert that to mass by dividing by 32 .2 feet per second square.

00:27 So for the mass, we get 0 .46 kilograms, and we're told that initially, at rest, the angle is 0 degrees.

00:38 So it starts when the ball is straight up.

00:42 So now let's think about the forces involved on the ball.

00:48 So if we're saying we want to find the angle when the compressive force is zero, the only force acting on the ball is the weight and the x direction.

00:58 So the weight in the x direction is mg cosine theta is equal to, newton's second law is force is equal to mass times the acceleration.

01:12 And here the acceleration experiences this intrepidal acceleration.

01:17 So what we want to find is this angle, angle theta.

01:21 We can expand this a little bit.

01:23 We can say m .g.

01:27 Cosine theta is equal to the mass.

01:30 And then this intrepidateal acceleration is a velocity squared divided by the radius of curvature.

01:37 And for this problem, this is the length of the right.

01:40 Which is three feet so now let's go find the velocity so since we're dealing with movement there has to be a kinetic energy involved and velocity is embedded in kinetic energy and the ball is at some height so we know that there's potential energy as well so you can use conservation of energy and that's the initial kinetic energy plus initial potential energy is equal to the final kinetic energy plus the final potential energy.

02:22 You know that initially the balls at rest when theta is equal to 0 degree, so the initial kinetic energy is 0.

02:29 And if we want to say at that rest point, we'll call that the initial height is 0, so then the potential energy is 0.

02:40 So then we're just left with final kinetic energy, 1 1 half m v2 squared, and then the potential energy is minus mgh2.

02:58 It's the movement to the right, the height gets smaller.

03:02 So it's a minus mgh2, and all that is equal to 0.

03:10 So to find the velocity, we need to find the height.

03:15 So if we think about this, initially the ball is straight up.

03:21 We'll call that point a.

03:22 And then it moves to the right, and then it moves to the right, and it reaches some angle when the compressive force is zero...

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