A factory worker pushes a $30.0$-kg crate a distance of $4.5$ m along a level floor at constant velocity by pushing horizontally on it. The coefficient of kinetic friction between the crate and the floor is $0.25$. How much work is done on the crate by this force?

Verified step by step guidance

1. Identify the forces acting on the crate: the pushing force (horizontal), the kinetic friction force opposing the motion, the gravitational force (weight), and the normal force from the floor.
2. Since the crate moves at constant velocity, the net force is zero. Therefore, the pushing force must be equal in magnitude to the kinetic friction force.
3. Calculate the normal force, which for a horizontal surface is equal to the weight of the crate: \(N = m \times g\), where \(m\) is the mass (30.0 kg) and \(g\) is the acceleration due to gravity (9.8 m/s\^2).
4. Calculate the kinetic friction force using \(f_k = \mu_k \times N\), where \(\mu_k\) is the coefficient of kinetic friction (0.25) and \(N\) is the normal force.
5. Calculate the work done by the pushing force using \(W = F \times d \times \cos(\theta)\), where \(F\) is the pushing force (equal to the friction force), \(d\) is the displacement (4.5 m), and \(\theta\) is the angle between the force and displacement (0 degrees since force is horizontal).

Key Concepts

Work

Work is defined as the product of the force applied to an object and the distance over which that force is applied, in the direction of the force. Mathematically, it is expressed as W = F × d × cos(θ), where W is work, F is the force, d is the distance, and θ is the angle between the force and the direction of motion. In this scenario, since the force is applied horizontally and the crate moves horizontally, θ is 0 degrees, simplifying the calculation to W = F × d.

Friction

Friction is a force that opposes the relative motion of two surfaces in contact. The coefficient of kinetic friction (μ_k) quantifies this force and is used to calculate the frictional force (F_friction) as F_friction = μ_k × N, where N is the normal force. In this case, the normal force equals the weight of the crate, which is the mass multiplied by gravitational acceleration, allowing us to determine the frictional force acting against the worker's push.

Constant Velocity

Constant velocity implies that an object moves at a steady speed in a straight line, meaning that its acceleration is zero. According to Newton's first law, if the crate is moving at constant velocity, the net force acting on it must be zero. Therefore, the force exerted by the worker must equal the frictional force opposing the motion, allowing us to calculate the work done by the worker in pushing the crate.