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. (a) What are the coefficients of static and kinetic friction between the crate and the floor?

Verified step by step guidance

1. Identify the forces acting on the crate. When the crate just begins to move, the force exerted (313 N) overcomes the static friction force. Once moving, the force required to maintain constant velocity (208 N) equals the kinetic friction force.
2. Calculate the coefficient of static friction (\(\mu_s\)). Use the formula \(\mu_s = \frac{F_{static}}{N}\), where \(F_{static}\) is the static friction force and \(N\) is the normal force. The normal force is equal to the weight of the crate, which can be calculated using \(N = mg\), where \(m\) is the mass of the crate and \(g\) is the acceleration due to gravity (approximately 9.8 m/s^2).
3. Calculate the coefficient of kinetic friction (\(\mu_k\)). Use the formula \(\mu_k = \frac{F_{kinetic}}{N}\), where \(F_{kinetic}\) is the kinetic friction force and \(N\) is the normal force, calculated the same way as in step 2.
4. Substitute the values into the formulas to find \(\mu_s\) and \(\mu_k\). For \(\mu_s\), use 313 N for \(F_{static}\) and for \(\mu_k\), use 208 N for \(F_{kinetic}\).
5. Interpret the results. The coefficients of static and kinetic friction tell you about the roughness and the nature of the interaction between the surfaces of the crate and the floor. A higher coefficient of static friction compared to kinetic friction is typical, as it usually takes more force to start moving an object than to keep it moving.

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 is multiplied by the normal force to determine 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 both static and kinetic friction. 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 the frictional forces acting on the crate.