To solve the problem, we need to follow these steps:
1. First, we are given the mass of the crate, which is 232 kg.
2. The angle made by the pulling force below the horizontal is also provided, which is 24 degrees.
3. Additionally, the problem provides the coefficient of kinetic friction, which is 0.299.
4. As the first step in solving the problem, we need to convert this angle from degrees to radians because most mathematical operations use radians as default.
5. After converting the angle to radians, the next step is to calculate the weight of the crate. Weight is calculated using the formula mass multiplied by gravity. So, the weight of the crate is 232 kg (mass) times 9.8 m/s² (gravity), which equals 2273.6 N.
6. With the weight of the crate known, we can then compute for the normal force, which is the force perpendicular (or "normal") to the surface the crate is on. Normal force is taken as the weight of the crate times the cosine of the angle, which results in approximately 2077.0369524962175 N.
7. The frictional force can be calculated next. Frictional force is derived from the relationship: Frictional force = Coefficient of Kinetic Friction * Normal force. Therefore it is 0.299(coefficient of kinetic friction) times 2077.0369524962175 N(normal force), which gives approximately 621.034048796369 N.
8. As per the problem, the net work done is zero. This means, the applied force should equal to the frictional force. As this applied force is making 24 degrees below the horizontal, the horizontal component of force is (P*cos24) which should equal the frictional force.
9. Rearranging this equation, we can solve for the pushing force (P), which boils down to dividing the frictional force by the cosine of the angle. Therefore, the pushing force equals to 621.034048796369 N(frictional force) divided by the cosine of 24 degrees angle, which results in approximately 679.8064 N.
In conclusion, you need to apply approximately 679.8064 N of force to ensure that the net work done by the pushing force and kinetic frictional force is zero.