A man drags a 180 lb crate across a floor by pulling on a rope inclined at 20.0 degrees with the horizontal. If the coefficient of static friction is 0.560 , and the coefficient of kinetic friction is 0.350, find the acceleration (in feet/second^2) of the crate at the instant it breaks loose and starts sliding.

no

To find the acceleration of the crate at the instant it breaks loose and starts sliding, we can use Newton's second law of motion:

1. Identify the forces acting on the crate:
- Tension force (T) in the rope, pulling the crate forward.
- Gravity force (Fg) acting vertically downward.
- Normal force (Fn) exerted by the floor on the crate.
- Friction force (Ff) acting parallel to the floor.

2. Calculate the force of gravity:
Fg = mass * acceleration due to gravity
= 180 lb * (32.2 ft/s^2) [1 lb = 32.2 ft/s^2]
= 5784 lb⋅ft/s^2

3. Calculate the applied force component by resolving the tension force:
T_parallel = T * sin(20.0 degrees)

4. Calculate the friction force (Ff) using the coefficient of static friction:
Ff = coefficient of static friction * Fn

5. To determine the maximum static friction force (Ff_max), we can use the equation:
Ff_max = coefficient of static friction * Normal force (Fn)
Since the upward normal force is equal to the downward gravitational force (Fn = Fg),
Ff_max = μ_static * Fg

6. When the crate starts sliding, the friction force is equal to the maximum static friction force, so:
Ff = Ff_max

7. Calculate the normal force (Fn):
Fn = Fg

8. Substitute the values into the equation Ff = coefficient of static friction * Fn and solve for Ff:
Ff = μ_static * Fn
Ff = 0.560 * Fn

9. Substitute the value of Ff into Newton's second law to calculate the net force:
Net force = T_parallel - Ff

10. Since the crate starts sliding, the maximum static friction force (Ff_max) changes to kinetic friction force (Fk), so:
Fk = coefficient of kinetic friction * Fn
Fk = 0.350 * Fn

11. Substitute the value of Fk into Newton's second law:
Net force = T_parallel - Fk

12. Equate the net force to the mass of the crate times its acceleration:
Net force = mass * acceleration

13. Rearrange the equation to solve for acceleration:
acceleration = (T_parallel - Fk) / mass

By plugging in the known values for T_parallel, Fk, and mass, we can calculate the acceleration of the crate at the instant it starts sliding.

To find the acceleration of the crate at the instant it breaks loose and starts sliding, we need to analyze the forces acting on the crate.

First, let's consider the forces in the horizontal direction. The force of gravity acting on the crate can be divided into two components: the component parallel to the incline and the component perpendicular to the incline. The force parallel to the incline is given by the weight of the crate (180 lb) multiplied by the sine of the angle of incline (20 degrees):

Force_parallel = 180 lb * sin(20 degrees)

The force of friction acting against the motion of the crate can be calculated using the coefficient of static friction and the normal force. The normal force is equal to the weight of the crate multiplied by the cosine of the angle of incline (since the crate is on an incline):

Normal_force = 180 lb * cos(20 degrees)

The force of static friction can be found by multiplying the coefficient of static friction by the normal force:

Force_static_friction = coefficient_static_friction * Normal_force

Since the crate is at the point of breaking loose and starting to slide, the force of static friction is equal to the maximum force of static friction, which is equal to the force of static friction just before the crate starts moving:

Force_static_friction = maximum_force_static_friction

So we can write:

maximum_force_static_friction = coefficient_static_friction * Normal_force

Now, once the crate breaks loose and starts sliding, the force of friction changes from static friction to kinetic friction. The force of kinetic friction can be calculated using the coefficient of kinetic friction and the normal force:

Force_kinetic_friction = coefficient_kinetic_friction * Normal_force

Since the crate has started sliding, the force of kinetic friction will oppose the motion of the crate. Therefore, we can write:

Force_kinetic_friction = -ma

where "m" is the mass of the crate and "a" is the acceleration of the crate.

To solve for the acceleration, we can set the force of static friction at the point of breaking loose equal to the force of kinetic friction:

maximum_force_static_friction = Force_kinetic_friction

Substituting the equations for the forces, we get:

coefficient_static_friction * Normal_force = coefficient_kinetic_friction * Normal_force

Since the normal force cancels out, we're left with:

coefficient_static_friction = coefficient_kinetic_friction

Finally, we can solve for the acceleration by rearranging the equation:

a = (maximum_force_static_friction) / m

To convert the answer to feet/second^2, we need to use the appropriate conversion factor.

Mass = 180Lb * 0.454kg/Lb = 81.72 kg.

m*g = 81.72 * 9.8 = 801 N = Wt. of crate
= Normal(Fn).

Fs = u*Fn = 0.56 * 801 = 449 N. = Force of static friction. = Hor. component of
applied force(Fx).

Fk = u*Fn = 0.350 * 801 = 280 N. = Force of kinetic friction.

a = (Fx-Fk)/m = (449-280)/81.72 = 2.07
m/s^2 = 6.82 Ft/s^2