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. (c) How much work is done on the crate by friction?

Verified step by step guidance

1. Identify the forces acting on the crate. Since the crate is moving at a constant velocity, the net force acting on it is zero. This means the force applied by the worker is equal in magnitude and opposite in direction to the force of friction.
2. Calculate the force of friction using the formula: \( F_{\text{friction}} = \mu_k \times F_{\text{normal}} \), where \( \mu_k \) is the coefficient of kinetic friction and \( F_{\text{normal}} \) is the normal force. For a horizontal surface, the normal force is equal to the weight of the crate, \( F_{\text{normal}} = m \times g \), where \( m \) is the mass of the crate and \( g \) is the acceleration due to gravity (approximately 9.8 m/s²).
3. Substitute the given values into the friction force formula: \( F_{\text{friction}} = 0.25 \times (30.0 \text{ kg} \times 9.8 \text{ m/s}^2) \).
4. Calculate the work done by friction using the formula: \( W_{\text{friction}} = F_{\text{friction}} \times d \times \cos(\theta) \), where \( d \) is the distance the crate is moved and \( \theta \) is the angle between the force of friction and the direction of motion. Since friction acts opposite to the direction of motion, \( \theta = 180^\circ \) and \( \cos(180^\circ) = -1 \).
5. Substitute the values into the work formula: \( W_{\text{friction}} = F_{\text{friction}} \times 4.5 \text{ m} \times (-1) \). This will give you the work done by friction on the crate.

Key Concepts

Work Done by a Force

Work is defined as the product of the force applied to an object and the distance over which the 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 magnitude of the force, d is the distance, and θ is the angle between the force and the direction of motion. In this scenario, the angle is 0 degrees since the force of friction and displacement are in the same direction.

Kinetic Friction

Kinetic friction is the force that opposes the motion of two surfaces sliding past each other. It is calculated using the formula f_k = μ_k * N, where f_k is the kinetic frictional force, μ_k is the coefficient of kinetic friction, and N is the normal force. For a horizontal surface, the normal force is equal to the weight of the object, which is the product of mass and gravitational acceleration (N = m * g).

Constant Velocity and Net Force

When an object moves at constant velocity, it implies that the net force acting on it is zero. This means that the applied force is equal and opposite to the frictional force. In this problem, since the crate is moving at constant velocity, the horizontal force applied by the worker is equal to the kinetic frictional force, ensuring no acceleration occurs.