An object of mass m is suspended from a post on top of a...

An object of mass $m$ is suspended from a post on top of a cart by a string of length $L$ as in Figure $\mathrm{P} 8.66 \mathrm{a} .$ The cart and object are initially moving to the right at constant speed $v_{i} .$ The cart comes to rest after colliding and sticking to a bumper as in Figure $\mathrm{P} 8.65 \mathrm{~b}$, and the suspended object swings through an angle $\theta .$ (a) Show that the speed is $v_{i}=\sqrt{2 g L(1-\cos \theta)}$. (b) If $L=1.20 \mathrm{~m}$ and $\theta=85.0^{\circ}$, find the initial speed of the cart. (Hint: The force exerted by the string on the object does no work on the object.)

An object of mass $m$ is suspended from a post on top of a cart by a string of length $L$ as in Figure $\mathrm{P} 8.66 \mathrm{a} .$ The cart and object are initially moving to the right at constant speed $v_{i} .$ The cart comes to rest after colliding and sticking to a bumper as in Figure $\mathrm{P} 8.65 \mathrm{~b}$, and the suspended object swings through an angle $\theta .$ (a) Show that the speed is $v_{i}=\sqrt{2 g L(1-\cos \theta)}$. (b) If $L=1.20 \mathrm{~m}$
and $\theta=85.0^{\circ}$, find the initial speed of the cart. (Hint:
The force exerted by the string on the object does no work on the object.)

Key Concepts

Conservation of Mechanical Energy

This concept states that in the absence of non-conservative forces doing work (like friction or air resistance), the total mechanical energy (the sum of kinetic and potential energies) in a system remains constant. In problems involving a swinging object or pendulum, the loss in kinetic energy is exactly balanced by the gain in gravitational potential energy, and vice versa.

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field. When an object is raised to a height, its GPE increases according to the relation ?PE = mgh, where h is the vertical displacement. In pendulum motion, the vertical displacement is related to the length of the pendulum and the cosine of the swing angle, which is essential for determining energy changes.

Pendulum Motion and Geometry

In pendulum motion, the swing of the object traces a circular arc. The change in height of the object as it swings through an angle can be determined geometrically using the pendulum length and the cosine of the angle. This relationship is crucial as it links the mechanical energy conservation with the angular displacement of the pendulum.

Work Done by Constraint Forces

Constraint forces, like the tension in the string of a pendulum, often do no work on an object because they act perpendicular to the displacement of the object. Recognizing that the tension does no work simplifies the analysis, allowing one to apply energy conservation principles directly between kinetic and gravitational potential energies.

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