A ball of mass m=300 g is connected by a strong string of length L=80.0 cm to a pivot and held i

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A ball of mass m=300 g is connected by a strong string of...

A ball of mass $m=300 \mathrm{g}$ is connected by a strong string of length $L=80.0 \mathrm{cm}$ to a pivot and held in place with the string vertical. A wind exerts constant force $F$ to the right on the ball as shown in Figure $\mathrm{P} 8.78$ . The ball is released from rest. The wind makes it swing up to attain maximum height $H$ above its starting point before it swings down again. (a) Find $H$ as a function of $F$ . Evaluate $H$ for $(b) F=$ 1.00 $\mathrm{N}$ and $(\mathrm{c}) F=10.0 \mathrm{N}$ . How does $H$ behave (d) as $F$ approaches zero and (e) as $F$ approaches infinity? (f) Now consider the equilibrium height of the ball with the wind blowing. Determine it as a function of $F$ . Evaluate the equilibrium height for $(g) F=10 \mathrm{N}$ and $(\mathrm{h})$ Fgoing to infinity.

Key Concepts

Force Equilibrium

Force equilibrium involves analyzing the balance between forces to determine a system's steady state or equilibrium position. In the context of a pendulum subjected to a horizontal force, equilibrium is reached when the net forces (including gravitational and external forces) balance such that the object remains stationary or in uniform motion, which helps ascertain the system's equilibrium height.

Asymptotic Behavior Analysis

Asymptotic analysis involves studying how a physical system behaves in the limits of certain parameters—in this case, as the force tends toward zero or infinity. Evaluating these limits helps to understand the response of the system under extreme conditions and to verify that the obtained relationships behave consistently with physical intuition.

Circular Motion Constraints

When an object is attached to a pivoted string, its motion is constrained to a circular arc. This geometric constraint relates the displacement in the horizontal direction to the vertical rise of the object. It is essential in determining how the work done by a horizontal force is converted into a vertical gain in gravitational potential energy, based on the angular displacement of the pendulum.

Gravitational Potential Energy

Gravitational potential energy is the energy associated with an object's position in a gravitational field. In problems involving changes in height, such as a pendulum swinging upward, the increase in gravitational potential energy (mgH) is a key factor since it must be provided by the work done by other forces (for example, the wind force).

Work-Energy Principle

This principle states that the net work performed on an object by external forces is equal to its change in kinetic energy. In scenarios where a constant external force acts on a moving object, such as the force from the wind in this problem, the work done by that force can be equated to the energy changes, typically converting kinetic energy into gravitational potential energy as the object rises.

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