a 314 kg crate is pulled by two horizontal forces. the first force is 255 n at a 43.5 angle and the second is 360 n at a 212 angle. what is the acceleration of the crate?

According to Newton's second law of motion, the net force (ΣF) on an object is equal to its mass (m) times its acceleration (a). By resolving the forces into x and y components, we can find the components of the acceleration vector. From there, we can use Pythagorean theorem to find the magnitude of the acceleration, and trigonometry to find the direction.

The x component of each force is F cos θ, where F is the magnitude of the force and θ is the direction. The net force in the x direction is therefore:

ΣFₓ = 255 N cos 43.5° + 360 N cos 212°

ΣFₓ = -120.3 N

The y component of each force is F sin θ, where F is the magnitude of the force and θ is the direction. The net force in the y direction is therefore:

ΣFᵧ = 255 N sin 43.5° + 360 N sin 212°

ΣFᵧ = -15.24 N

Using Newton's second law of motion, the components of the acceleration are:

ΣF = ma

a = ΣF / m

aₓ = -120.3 N / 314 kg = -0.383 m/s²

aᵧ = -15.24 N / 314 kg = -0.049 m/s²

Using Pythagorean theorem, the magnitude of the acceleration is:

a² = aₓ² + aᵧ²

a² = (-0.383 m/s²)² + (-0.049 m/s²)²

a = 0.386 m/s²

Using trigonometry, the direction of the acceleration is:

tan θ = aᵧ / aₓ

tan θ = -0.049 m/s² / -0.383 m/s²

θ = 187°