(III) Two masses, m1=18.0 kg and m2=26.5 kg , are connected by a rope that hangs over a pulley (

step 1 Question number 49, when mass M2 drops, then its initial gravitational potential energy is M2GH.

(III) Two masses, m1=18.0 kg and m2=26.5 kg , are connected...

(III) Two masses, $m_{1}=18.0 \mathrm{kg}$ and $m_{2}=26.5 \mathrm{kg}$ , are connected by a rope that hangs over a pulley (as in Fig. $8-47$ ). The pulley is a uniform cylinder of radius 0.260 $\mathrm{m}$ and mass 7.50 $\mathrm{kg}$ . Initially, $m_{1}$ is on the ground and $m_{2}$ rests 3.00 $\mathrm{m}$ above the ground. If the system is now released, use conservation of energy to determine the speed of $m_{2}$ just before it strikes the ground. Assume the pulley is frictionless.

(III) Two masses, $m_{1}=18.0 \mathrm{kg}$ and $m_{2}=26.5 \mathrm{kg}$ , are connected by a rope that hangs over a pulley (as in Fig. $8-47$ ). The pulley is a uniform cylinder of radius 0.260 $\mathrm{m}$ and mass 7.50 $\mathrm{kg}$ . Initially, $m_{1}$ is on the ground and $m_{2}$ rests 3.00 $\mathrm{m}$ above the ground. If the system is now released, use conservation of energy to determine
the speed of $m_{2}$ just before it strikes the ground. Assume the pulley is frictionless.

Key Concepts

Conservation of Energy

This principle states that the total mechanical energy of an isolated system remains constant if only conservative forces act on it. In problems like this, it is applied by equating the initial total energy (including potential energy) to the final total energy (including kinetic energy components) to solve for unknown quantities such as speed.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field, typically expressed as mgh. In the context of connecting masses and pulleys, differences in gravitational potential energy are converted into kinetic energy as the masses move under gravity.

Translational Kinetic Energy

Translational kinetic energy is the energy an object possesses due to its motion in a straight line and is calculated using ½mv². In problems involving connected masses, the translational kinetic energies of the moving masses are key components in the energy balance.

Rotational Kinetic Energy

Rotational kinetic energy is the energy due to the rotation of an object and is given by ½I?², where I is the moment of inertia and ? is the angular speed. In systems with pulleys, the rotational kinetic energy of the pulley must be included when applying conservation of energy.

Moment of Inertia

The moment of inertia is a measure of an object’s resistance to changes in its rotational motion. For a uniform cylinder, it quantifies how mass is distributed relative to the axis of rotation and is essential for calculating the rotational kinetic energy in pulley problems.

Relationship between Linear and Angular Speed

This concept links the linear speed of a point on the rim of a rotating object to its angular speed by the relation v = r?. It is critical in pulley systems to relate the rotation of the pulley to the translational motion of the connected masses.

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