In Fig. 6-62, a 1.0 kg pencil box on a 30^∘ frictionless incline is connected to a 3.0 kg pen

step 1 So we've got this mass here, which is a pencil box. And we are connecting to a pen box up here.

In Fig. 6-62, a 1.0 kg pencil box on a 30^∘ frictionless...

In Fig. 6-62, a $1.0 \mathrm{~kg}$ pencil box on a $30^{\circ}$ frictionless incline is connected to a $3.0 \mathrm{~kg}$ pen box on a horizontal frictionless $\quad$ surface. The pulley is frictionless and massless. (a) If the magnitude of the applied force $F$ is $2.3 \mathrm{~N}$, what is the tension in the connecting cord? (b) What is the largest value that the magnitude of $\vec{F}$ may have without the connecting cord becoming slack?

Key Concepts

Forces on an Inclined Plane

When an object is placed on an inclined plane, gravitational force is split into components: one parallel to the plane which causes motion, and one perpendicular which is balanced by the normal force. Analyzing these components is key to understanding the dynamics of objects on slopes and is essential in problems involving inclined surfaces.

Frictionless Surfaces

Frictionless surfaces are an idealization in physics where frictional forces are assumed to be negligible or zero. This assumption simplifies the problem by eliminating kinetic friction from the equations of motion, allowing a clearer focus on other forces such as gravity, applied forces, and tension.

Pulley Systems and Constraints

Pulley systems are used to change the direction of an applied force and can also affect the magnitude of the force in mechanical advantage scenarios. The constraints imposed by the inextensible rope ensure that connected objects have related accelerations, thus linking the motion of different bodies and aiding in the formulation of equations to analyze the system.

Free-Body Diagrams

Free-body diagrams are graphical representations used to isolate a single body and depict all the external forces acting on it. They simplify the process of applying Newton's laws by visually organizing forces such as gravity, normal force, tension, and applied forces, which is crucial for understanding and solving problems in mechanics.

Newton's Second Law

This fundamental principle of mechanics states that the acceleration of an object is directly proportional to the net force acting upon it and inversely proportional to its mass. It is used to calculate the net forces on bodies in motion by setting up equations that relate force, mass, and acceleration, which is essential when analyzing connected systems.

Tension in a Rope

Tension refers to the pulling force transmitted along a rope, string, or cable when it is subjected to forces at its ends, especially in systems with pulleys. In idealized cases where the rope is massless and the pulley is frictionless, the tension remains constant throughout the rope, which simplifies the analysis of the system.

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