Il nGues pr. 39 shows a block of mass m resting on a 20^∘...

Il nGues pr. 39 shows a block of mass $m$ resting on a $20^{\circ}$ slope. The block has coefficients of friction $\mu_{s}=0.80$ and $\mu_{\mathrm{k}}=0.50$ with the surface. It is connected via a massless string over a massless, frictionless pulley to a hanging block of mass $2.0 \mathrm{kg}$ a. What is the minimum mass $m$ that will stick and not slip? b. If this minimum mass is nudged ever so slightly, it will start being pulled up the incline. What acceleration will it have?

Il nGues pr. 39 shows a block of mass $m$ resting on a $20^{\circ}$ slope. The block has coefficients of friction $\mu_{s}=0.80$ and $\mu_{\mathrm{k}}=0.50$ with the surface. It is connected via a massless string over a massless, frictionless pulley to a hanging block of mass $2.0 \mathrm{kg}$
a. What is the minimum mass $m$ that will stick and not slip?
b. If this minimum mass is nudged ever so slightly, it will start being pulled up the incline. What acceleration will it have?

Key Concepts

Newton's Second Law

Newton's second law is a fundamental principle that relates the net force acting on an object to its mass and the resulting acceleration (F = m·a). This law is applied in problems involving dynamics to set up equations that describe how forces like gravity, friction, and tension collectively determine the motion of the object.

Tension in Massless Strings and Frictionless Pulleys

In systems involving massless strings and frictionless pulleys, the tension is assumed to be uniform throughout the string. This simplifies the analysis because forces can be equated across the system without accounting for mass or frictional losses in the string and pulley, allowing for direct application of Newton's laws to each connected mass.

Kinetic Friction

Kinetic friction comes into play once an object is in motion. It acts to oppose the movement and is generally lower than the maximum static friction. The force of kinetic friction is calculated as the product of the kinetic friction coefficient and the normal force, and it is used in determining the acceleration of objects once they start moving.

Components of Gravitational Force on an Incline

When an object is on an inclined plane, the gravitational force acting on it can be decomposed into two perpendicular components: one parallel to the slope and one perpendicular to it. The parallel component drives the object down the incline (m·g·sin?), while the perpendicular component (m·g·cos?) influences the normal force, which in turn affects frictional forces.

Free-Body Diagrams

A free-body diagram is a graphical representation of an object and all the external forces acting upon it. It is a fundamental tool used in mechanics to isolate a body and identify the forces such as gravity, friction, normal force, and tension. This aids in setting up equations based on Newton's laws to analyze the motion or equilibrium of the system.

Static Friction

Static friction is the force that opposes the initiation of motion between two surfaces in contact. It acts up to a maximum value given by the product of the static friction coefficient and the normal force. Determining the threshold at which static friction is overcome is crucial for solving problems about when an object begins to move.

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