The 1-kg ball at A is suspended by an inextensible cord and given an initial horizontal velocity

The 1-kg ball at A is suspended by an inextensible cord and...

Key Concepts

Projectile Motion

This concept involves analyzing the trajectory of an object moving under the influence of gravity alone after it is set in motion. The problem requires determining the vertical displacement of the ball when it reaches a certain horizontal position, so understanding the time of flight, the initial velocity components, and how gravity acts on the object are key elements in solving the problem.

Conservation of Energy

In problems like these, energy conservation is used to relate the initial kinetic energy given to the ball to its potential energy at different points along its trajectory. This principle allows one to determine the speed at various positions during the motion, especially when the object’s gravitational potential energy changes as it swings along a curved path.

Circular Motion and Pendulum Dynamics

Since the ball is initially attached to an inextensible cord, it follows a circular arc, which is a key aspect of pendulum motion. Analyzing the forces and velocity components in circular motion, including centripetal acceleration, helps determine the conditions at the release point, which then influences the projectile motion analysis.

Inextensible Cord Constraint

The inextensibility of the cord imposes a fixed radius for the circular path of the ball. This constraint is vital for the geometric analysis of the problem because it sets the relationship between the ball’s position and the pivot point, thereby affecting both the energy conservation and the transition from circular to projectile motion.

Transcript

00:01 So here we have a ball hanging from a cord up here and then there's a basket up here.

00:08 And we want to know the basket is actually right up directly above this point because we're told that xp is zero.

00:15 So the basket is directly above this point.

00:18 And we're told that we asked how what initial velocity much to give this ball so that it comes up here and deposits itself in the basket.

00:30 And the big question then is, is when does this cable here or this cord or rope become slack? because after it becomes slack, then the ball is kind of being freefall.

00:45 So that's the first thing we need to figure out is what angle, given a velocity, what angle does the ball, does the rope here become slack? so if it's taught, we know that it's just kind of the equations for a pendulum.

01:07 So if we measure kinetic potential energy from this point, initially we're at minus mgl, where l is the length of this chord, and plus 1 half mv not squared.

01:18 And then at some arbitrary point up here at theta, we're at kinetic energy of 1 half mv squared and a potential energy of mgl sine theta.

01:31 Now if we draw a 3x diagram of the particle, of the ball here, we have the weight and we have the tension in the cable.

01:39 Tension in the cable is acting in the, as i've drawn it, in the negative radial direction, so that's this.

01:46 The mass of the weight of the ball is acting in the negative horizontal direction, or vertical direction.

01:55 And then the acceleration is acting in the negative radial direction.

02:02 And so that's mass times acceleration is minus one half mv squared over l in the radial direction.

02:11 And so now if we take the dot product of this thing with the radial unit vector, take the dot product of all that, and we'll see, we arrange some things.

02:20 We get the t equals m times the quantity v0 squared all over l, minus g times quantity 2 plus 3 sine theta.

02:31 And again, i've substituted in here, i've solved up here for v squared, and i substituted that in and done some simplifications to get this equation here for t.

02:49 And so we can figure out if t when t equals zero, we can set t to zero and we can figure out with the theta when that happens.

02:59 And so we're actually given, it's a little easier here because we're not actually asked what the initial velocity be.

03:07 We're actually given it and asked if it will determine the height here of the basket such that it will drop in.

03:16 So we're given this initial velocity so we can figure out where the angle is that this rope becomes slack and that's 48 .5 degrees.

03:26 So a little past 45 degrees, it becomes slack.

03:34 And then at that point, the velocity is going to be somewhere, you know, in the, in the, oh, sorry, we're measuring theta from zero.

03:47 That makes sense because things weren't working out, right? so we're measuring theta positive from this angle...

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