A ball swings in a vertical circle at the end of a rope 1.50...

A ball swings in a vertical circle at the end of a rope $1.50 \mathrm{~m}$ long. When the ball is $36.9^{\circ}$ past the lowest point and on its way up, its total acceleration is $(-22.5 \mathbf{i}+$ $20.2 \mathrm{j}) \mathrm{m} / \mathrm{s}^{2}$. At that instant, (a) sketch a vector diagram showing the components of this acceleration, (b) determine the magnitude of its radial acceleration, and (c) determine the speed and velocity of the ball.

Key Concepts

Centripetal (Radial) Acceleration

This concept refers to the component of acceleration that is directed toward the center of the circle along which an object is moving. It arises in all types of circular motion and keeps the object moving in its circular path by changing the direction of the velocity vector. Its magnitude is given by the relation a_r = v^2/r, where v is the speed and r is the radius of the circle.

Tangential Acceleration

Tangential acceleration is the component of an object's acceleration that is tangent to the circular path. It is responsible for changes in the magnitude of the object's velocity (speeding up or slowing down) along the path. Unlike centripetal acceleration, tangential acceleration does not affect the direction of the velocity vector, only its magnitude.

Vector Decomposition

Vector decomposition involves breaking down a vector into its component parts along specific directions, such as horizontal and vertical, or radial and tangential directions. In the context of circular motion, decomposing the total acceleration into radial and tangential components helps in analyzing how the forces and accelerations contribute to the motion and in understanding the dynamics of the system.

Kinematics of Circular Motion

This concept covers the principles and equations that describe the motion of an object moving along a circular path. It involves understanding how the radial (centripetal) and tangential components of acceleration relate to the object's speed and the curvature of its path. Kinematics of circular motion is essential for solving problems that involve variable speed and complex motion paths, including determining the instantaneous velocity and acceleration vectors.

Transcript

0:00 Hello.

00:02 In this problem, we have a ball that is swinging in a vertical circle, and we are given the coordinates or in vector notation the total acceleration of this ball, and it's negative 22 .5 in the i direction and plus 20 .2 in the j's direction.

00:22 We're asked to draw a vector notation or a vector diagram of this acceleration and show each component.

00:29 I did it here.

00:30 In this drawing and i did some extra stuff jet that will be helpful in the next parts b and c.

00:38 So as you can see here, the vectors shown in blue.

00:42 This is the components of the acceleration as given, knowing that the i direction is to the right.

00:48 Here is the negative i, and this is the negative j direction, and this is the j direction.

00:54 Of course, all of them are unit vectors.

00:57 What about the ones in magenta? the ones in magenta are the radial basis or the radial and tangential basis.

01:04 This is the radial and this is the tangential and they are valid bases just as the i and j basis.

01:10 So in part b, actually, we are asked to calculate the radial acceleration.

01:18 So how to get the radial acceleration? basically, we need to these two components and see how much of each of these is projected on this magenta.

01:29 Arrow here that points to the center.

01:32 So this theta that it makes with the vertical, of course, it's 36 .9 as shown in the problem...

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