A pendulum of length 1.00 m swings in a vertical plane...

A pendulum of length $1.00 \mathrm{~m}$ swings in a vertical plane (Fig. $4.19$ ). When the pendulum is in the two horizontal positions $\theta=90^{\circ}$ and $\theta=270^{\circ}$, its speed is $5.00 \mathrm{~m} / \mathrm{s}$. (a) Find the magnitude of the radial acceleration and tangential acceleration for these positions. (b) Drawa vector diagram to determine the direction of the total acceleration for these two positions. (c) Calculate the magnitude and direction of the total acceleration.

A pendulum of length $1.00 \mathrm{~m}$ swings in a vertical plane (Fig. $4.19$ ). When the pendulum is in the two horizontal positions $\theta=90^{\circ}$ and $\theta=270^{\circ}$, its speed is $5.00 \mathrm{~m} / \mathrm{s}$.
(a) Find the magnitude of the radial acceleration and tangential acceleration for these positions. (b) Drawa vector diagram to determine the direction of the total acceleration for these two positions. (c) Calculate the magnitude and direction of the total acceleration.

Key Concepts

Centripetal (Radial) Acceleration

In circular motion, any object moving along a curved path must accelerate toward the center of the circle. This inward acceleration, called centripetal acceleration, is given by the formula a?rad? = v²/r. It is responsible for changing the direction of the velocity vector, keeping the object on its curved path, even if the speed remains constant.

Tangential Acceleration

Tangential acceleration is the component of acceleration that acts along the direction of the object's velocity, affecting the speed of the object along the circular path. In systems where other forces (like gravity) act, tangential acceleration results from the net force component parallel to the path, and can be computed by considering the projection of forces in the direction tangential to the circle.

Decomposition of Forces and Free Body Diagrams

In problems involving pendulum motion, the forces acting on the bob are typically resolved into components that are radial (pointing towards the pivot) and tangential (along the direction of motion). Drawing a free body diagram and decomposing forces helps determine the contributions to centripetal and tangential accelerations, such as extracting the component of gravitational force that acts along the arc of the pendulum's path.

Vector Addition of Perpendicular Components

When an object has acceleration components in perpendicular directions, such as the inward (radial) and along-the-path (tangential) directions in circular motion, the total acceleration can be determined by vector addition. This involves finding the magnitude using the Pythagorean theorem and obtaining the direction by taking the inverse tangent of the ratio of the components.

Transcript

0:00 Hello.

00:01 In this problem, we have a pendulum that is one meter in length, and we are given two positions at 90 degrees, given that theta is calculated from the vertical here.

00:15 So at 90 degrees, which is at this point, let me make it in red, at this point, and at 270, which is at this other point here, those are the two horizontal positions.

00:33 It says that at these two points, the velocity of this particle is 5 meters per second.

00:43 Here it's 5 meters per second as well.

00:46 And we are asked to calculate part a, the radial acceleration, and the tangential acceleration at these two positions.

00:56 So we know that the radial acceleration can be calculated directly from the equation of v squared divided by r.

01:04 So it's pretty straightforward here.

01:07 V is 5 meters per second.

01:09 So v squared is 25 and divided by r, which is 1 meter.

01:14 So we have the radial acceleration of 25 meters per second square.

01:22 For the tangential acceleration, the only force or the only kind of force that is acting here on this particle or on the pendulum, other than the centributal acceleration is the gravity.

01:36 Itself because the pendulum is a swinging in the vertical plane.

01:40 So here, the gravitational acceleration always points downward, and we can see that at this point, this is actually the tangential acceleration, because pointing downward, that's the tangent of the circle at this point.

01:58 So the tangential acceleration is just the gravitational acceleration, which is 9 .8 meters per second square.

02:05 And part b were asked to draw a free body diagram of the pendulum and draw the acceleration or the components of the acceleration.

02:19 So here we have at this position, which is 270 degrees from the vertical...

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