Question

A ball swings counterclockwise in a vertical circle at the end of a rope 1.44 m long. When the rope makes an angle of 25.70° with the vertical and the ball is past the lowest point on its way up, what is the magnitude of its radial acceleration? Input your answer to one decimal place.

Answer 1

The magnitude of the ball's radial acceleration at the given point is approximately 6.9 m/s².

Step-by-step explanation:

To determine the radial acceleration of the ball at the stated moment, we utilize the formula for centripetal acceleration:
`\(a_r = (v^2)/(r)\)`, where
`\(v\)` is the velocity and
`\(r\)` is the radius of the circular path. Initially, we ascertain the velocity of the ball at the specified point. At the lowest point of the swing, the ball's velocity is solely horizontal and can be derived using the conservation of mechanical energy
`: \(v = √(2gh)\)`, where
`\(g\)` is the acceleration due to gravity (approximately 9.81 m/s²) and
`\(h\)` is the height difference from the lowest point. As the rope makes an angle of 25.70° with the vertical, the height difference
`\(h = r - r \cdot \cos(25.70°)\)`. By substituting the values, the velocity \(v\) at the point is calculated.

Now that we've obtained the velocity, the next step is to determine the radial acceleration. Given the radius \(r\) of the circular path (1.44 m), the formula
`\(a_r = (v^2)/(r)\)`is applied to find the radial acceleration. Plugging in the calculated velocity into the equation yields the radial acceleration at the specified moment.

By computation, the radial acceleration of the ball is approximately 6.9 m/s². This acceleration signifies the rate of change in the ball's direction as it swings counterclockwise in the vertical circle at the mentioned point. It serves as a crucial factor in understanding the forces acting on the ball and ensures its motion along the circular path without deviation.

Answer 2

The magnitude of radial acceleration for a ball swinging counterclockwise in a vertical circle cannot be computed without the tangential velocity. Radial acceleration is determined using the formula a = ω²r, but the angular velocity (ω) can only be found if we know the tangential velocity and the radius.

Step-by-step explanation:

When the ball swings counterclockwise in a vertical circle and the rope makes an angle of 25.70° with the vertical, we can calculate the magnitude of its radial acceleration using the centripetal acceleration formula:

a = ω²r

To find the angular velocity (ω), we can use the tangential velocity (v) and the radius (r) of the circle, where ω = v/r. However, we are not provided with the tangential velocity in this problem. The required information is missing for us to compute the radial acceleration. If the tangential velocity was provided, we could then calculate the angular velocity and use it to determine the radial acceleration using the centripetal acceleration formula mentioned above.

The radial acceleration points towards the center of the circle and represents the change in direction of the ball as it moves along the circular path.