Calculate Acceleration of object in Circular motion

[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

Homework Equations

The Attempt at a Solution

 

[Astronuc]

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.

 

[thomate1]

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.