A small lead ball, attached to a 0.75-m rope, is being...

A small lead ball, attached to a 0.75-m rope, is being whirled in a circle that lies in the vertical plane. The ball is whirled at a constant rate of three revolutions per second and is released on the upward part of the circular motion when it is $1.5 \mathrm{m}$ above the ground. The ball travels straight upward. In the absence of air resistance, to what maximum height above the ground does the ball rise?

Key Concepts

Uniform Circular Motion

Uniform circular motion refers to motion in a circular path at a constant speed. The key aspects are the radius of the circle and the constant angular speed, which combines to give a constant tangential velocity. This concept is crucial for analyzing the motion before the ball is released, because it provides the framework for determining the speed at which the ball is moving when it leaves the circular path.

Angular Speed and Tangential Velocity

Angular speed is the rate at which an object rotates around a circle and is typically measured in radians per second. Tangential velocity is the linear speed of the object moving along the circular path, and it is obtained by multiplying the angular speed by the radius. Understanding this relationship is essential because it allows one to calculate the initial speed of the ball as it transitions from circular motion to projectile motion.

Projectile Motion

Projectile motion describes the trajectory of an object that is launched into the air and moves under the influence of gravity alone. Once the ball is released from the circular path, it follows a projectile motion with an initial velocity and then decelerates upward due to gravity until it comes to rest momentarily at its maximum height. This concept is key for determining the highest altitude reached after the release.

Conservation of Energy in Projectile Motion

The principle of conservation of energy states that the total mechanical energy of a system remains constant if only conservative forces are acting. In the context of projectile motion, the kinetic energy the ball possesses at the moment of release is gradually converted into gravitational potential energy as it climbs. This conversion allows us to calculate the additional height the ball attains after release by equating the initial kinetic energy to the gravitational potential energy at the highest point.

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