A $45.0$-kg crate of tools rests on a horizontal floor. You exert a gradually increasing horizontal push on it, and the crate just begins to move when your force exceeds $313$ N. Then you must reduce your push to $208$ N to keep it moving at a steady $25.0$ cm/s. What are the coefficients of static and kinetic friction between the crate and the floor?

Verified step by step guidance

1. Step 1: Understand the problem. The problem involves determining the coefficients of static and kinetic friction. The coefficient of static friction (μ_s) is related to the force required to start moving the crate, while the coefficient of kinetic friction (μ_k) is related to the force required to keep the crate moving at a constant velocity.
2. Step 2: Write the formula for the force of friction. The force of friction is given by the equation: $f=\mu N$, where $\mu$ is the coefficient of friction and $N$ is the normal force. For a horizontal surface, the normal force is equal to the weight of the crate: $N=mg$, where $m$ is the mass of the crate and $g$ is the acceleration due to gravity.
3. Step 3: Calculate the coefficient of static friction. The force required to overcome static friction is given as 313 N. Using the formula for static friction: $f=\mu N$, substitute $f=313$ N and $N=mg=45.0$ kg $\cdot 9.8$ m/s². Solve for $\mu s_{}$.
4. Step 4: Calculate the coefficient of kinetic friction. The force required to keep the crate moving at a constant velocity is given as 208 N. Using the formula for kinetic friction: $f=\mu k_{}N$, substitute $f=208$ N and $N=mg=45.0$ kg $\cdot 9.8$ m/s². Solve for $\mu k_{}$.
5. Step 5: Interpret the results. The coefficients of static and kinetic friction are dimensionless quantities that describe the interaction between the crate and the floor. A higher coefficient of static friction indicates that more force is required to start moving the crate, while a lower coefficient of kinetic friction indicates that less force is required to keep it moving at a constant velocity.

Key Concepts

Static Friction

Static friction is the force that must be overcome to start moving an object at rest. It acts in the opposite direction of the applied force and varies up to a maximum value, which is determined by the coefficient of static friction and the normal force. In this scenario, the maximum static friction force is equal to the applied force just before the crate begins to move.

Kinetic Friction

Kinetic friction is the force that opposes the motion of two surfaces sliding past each other. It is generally less than static friction and is characterized by the coefficient of kinetic friction, which, when multiplied by the normal force, gives the frictional force acting on a moving object. In this case, the kinetic friction force is what allows the crate to move at a constant speed once it is in motion.

Normal Force

The normal force is the perpendicular force exerted by a surface to support the weight of an object resting on it. It acts against gravity and is crucial for calculating frictional forces. In this problem, the normal force is equal to the weight of the crate, which is the product of its mass and the acceleration due to gravity, and it directly influences both static and kinetic friction.