Kinetics, force and acceleration

[red123]

Homework Statement

The crate has a mass of 80 kg and is being towed by a chain which is always directed at 20º from the horizontal has shown. Determine the crate's acceleration in t = 2 s if the coefficient of static friction is μ_s = 0.4, the coefficient of kinetic friction is μ_k = 0.3, and the towing force is P = (90t²) N, where t is in seconds.

Homework Equations

ΣF = ma

The Attempt at a Solution

ΣF_x = ma_x
Pcos(20º) - μ_kmg = ma
84.57t² - 235.44 = (80 kg)a

solving for a when t = 2
a = 1.29 m/s²

The answer at the back of my book is a = 1.75 m/s². I think I'm supposed to consider that static friction force is greater than kinetic friction force, but I can't figure out how to do that.

 

[Doc Al]

ΣF_x = ma_x
Pcos(20º) - μ_kmg = ma

The force of friction is μN, but N does not simply equal mg. Solve for N, taking into consideration that the applied force is at an angle.

 

[red123]

The force of friction is μN, but N does not simply equal mg. Solve for N, taking into consideration that the applied force is at an angle.

Then N = mg*cos(20°)?
But that gives a = 1.46 m/s², not a = 1.75 m/s².

 

[Doc Al]

Then N = mg*cos(20°)?

How did you come to that conclusion?

To solve for N, add up the vertical components of all forces acting on the body. (N is one of those forces.) The sum must be zero, since the body is not accelerating vertically.

 

[red123]

Thank you, Doc Al.