Problem Statement:
A man is pushing a 60lb crate with a force F that is always directed down at 30 degrees from the horizontal. The force magnitude is increased until the crate begins to slide. We need to determine the initial acceleration of the crate given that the static coefficient of friction is 0.6 and the kinetic coefficient of friction is 0.3.

Solution:

1. Identifying the Forces:
To determine the initial acceleration of the crate, we first need to identify the forces acting on it. These forces include:
- Weight: The weight of the crate acts vertically downward and can be calculated using the formula: Weight = mass x acceleration due to gravity. In this case, the weight is 60lb.
- Normal Force: The normal force acts perpendicular to the surface of contact between the crate and the floor. It counterbalances the weight of the crate and can be calculated as Normal Force = Weight x cos(θ), where θ is the angle between the weight and the horizontal. In this case, θ is 30 degrees.
- Frictional Force: The frictional force opposes the motion of the crate and can be calculated as Frictional Force = coefficient of friction x Normal Force. Here, we have two types of coefficients of friction:
- Static Coefficient of Friction (μs): This is the coefficient of friction when the crate is at rest and not moving. In this case, μs is 0.6.
- Kinetic Coefficient of Friction (μk): This is the coefficient of friction when the crate is in motion. In this case, μk is 0.3.
- Applied Force: The man is applying a force F on the crate, which is directed down at 30 degrees from the horizontal.

2. Analyzing the Forces:
- When the crate is at rest, the static frictional force (Fs) opposes the applied force and prevents the crate from sliding. We can calculate the maximum static frictional force using the formula Fs = μs x Normal Force.
- When the applied force (F) exceeds the maximum static frictional force (Fs), the crate starts to slide. At this point, the static frictional force changes to the kinetic frictional force (Fk), which is given by Fk = μk x Normal Force.
- The net force acting on the crate is the difference between the applied force and the kinetic frictional force: Net Force = F - Fk.
- Using Newton's second law of motion (F = m x a), where F is the net force and m is the mass of the crate, we can calculate the acceleration (a) of the crate.

3. Calculating the Initial Acceleration:
To calculate the initial acceleration of the crate, we need to compare the applied force (F) with the maximum static frictional force (Fs).
- If F ≤ Fs, the crate remains at rest, and the acceleration is zero.
- If F > Fs, the crate starts to slide, and the acceleration is given by a = (F - Fk) / m.

4. Final Calculation:
- First, we calculate the maximum static

My answer is 0.75 m/s^2.