Energy in Connected Objects (Systems) Practice Problems

A cargo sled with a mass of $4.0$ kg, made from a durable composite material, is positioned on a frictionless aluminum incline at a $30$-degree angle. It is connected through a smooth, lightweight pulley to a vertically hanging steel crate that weighs $6.0$ kg. The crate starts $1.0$ meter above the ground. Find the speed of the sled and crate just before the crate hits the ground using energy conservation methods.

In a storage room, a heavy box of mass $10.0$ kg is initially at rest on a spring with a spring constant of $800$ N/m. The box is part of a pulley system where it is connected to a $3.0$-kg package. When the system is released, the heavier box compresses the spring by moving downwards, pulling the lighter package upwards. Calculate the distance the box falls, compressing the spring, before the entire system momentarily comes to rest.