# Calculate Acceleration of object in Circular motion

• Jordash
In summary, the conversation discusses the conical pendulum and its motion represented in a figure. The length of the string, angle, and time of one revolution are given. The task is to find the acceleration by deriving a symbolic equation. The homework equations and attempt at a solution are also mentioned, along with a suggestion to refer to a specific resource for further information.
Jordash

## Homework Statement

A conical pendulum is made of a ball on the end of a string, moving in circular motion as represented in the ﬁgure below. The length L of the string is 1.70 m, the angle θ is 37.0o, and the ball completes one revolution every 2.30 s

Calculate the acceleration of the ball, by ﬁrst deriving a symbolic equation

## The Attempt at a Solution

Please show some effort and work.

The ball is moving in a circular motion. Find the radius from the length of the string from pivot to ball, and the angle.

Acceleration is simply the change in velocity with respect to time.

Please refer to - http://hyperphysics.phy-astr.gsu.edu/hbase/rotq.html#rq - as well as one's text.

To calculate the acceleration of the ball in this conical pendulum system, we can use the equation for centripetal acceleration: a = v^2/r, where v is the velocity of the ball and r is the radius of the circular motion. Since the ball is moving in a conical motion, we can break down the velocity into its horizontal and vertical components, and use trigonometric functions to relate them to the angle θ and the length L of the string.

Thus, the horizontal component of the velocity is v_h = v*cosθ, and the vertical component is v_v = v*sinθ. We can then use the Pythagorean theorem to find the total velocity v: v^2 = v_h^2 + v_v^2.

Next, we can substitute the value for v into the equation for centripetal acceleration, giving us a = (v^2/r) = (v_h^2 + v_v^2)/r. Using the values given in the problem, we can plug in v_h = v*cos37.0o, v_v = v*sin37.0o, and r = L = 1.70 m.

Simplifying the equation, we get a = v^2/L = (v^2*cos^2θ + v^2*sin^2θ)/L = v^2*(cos^2θ + sin^2θ)/L. Since cos^2θ + sin^2θ = 1, the equation becomes a = v^2/L.

Finally, we can solve for v by using the formula for angular velocity: ω = 2π/T, where T is the period of the ball's motion. In this case, the period is given as 2.30 s, so ω = 2π/2.30 = 2.73 rad/s. Plugging this into the formula for v, we get v = ω*r = 2.73*1.70 = 4.64 m/s.

Substituting this value for v into our equation for acceleration, we get a = (4.64^2)/1.70 = 12.66 m/s^2. Therefore, the acceleration of the ball in this conical pendulum system is 12.66 m/s^2.

## 1. What is the formula for calculating acceleration in circular motion?

The formula for calculating acceleration in circular motion is a = v^2/r, where a is acceleration, v is velocity, and r is the radius of the circular path.

## 2. How is centripetal acceleration different from tangential acceleration?

Centripetal acceleration refers to the acceleration of an object towards the center of a circular path, while tangential acceleration refers to the change in an object's velocity in the direction tangential to the circular path.

## 3. How does the speed of an object in circular motion affect its acceleration?

The speed of an object in circular motion is directly proportional to its acceleration. This means that as the speed increases, the acceleration also increases, and vice versa.

## 4. Can an object have constant speed but changing acceleration in circular motion?

Yes, an object can have a constant speed but changing acceleration in circular motion. This occurs when the direction of the object's velocity changes, causing a change in its acceleration, but the magnitude of its velocity remains the same.

## 5. How does the radius of the circular path affect an object's acceleration?

The radius of the circular path is inversely proportional to an object's acceleration. This means that as the radius increases, the acceleration decreases, and vice versa. This is because a larger radius results in a longer distance to travel in the same amount of time, causing a lower acceleration.

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