I keep on getting the wrong answer I don't know if there is even static friciton in this problem in the begining before the rope snaps or not and if so if it must be equal to the Mu they give because there is only one but you can't have the same values for both coffecients of friction so I don't know how to do this problem

A 25-kg crate is held stationary on an incline by a rope with a tension of 32 N. Suddenly the rope snaps, sending the crate into a downard slide. If the coefficent of friction between the crate and the incline is .73, how many seconds will it take fo rthe crate to reach a speed of 5.0 m/s?

Answer from back of book 3.9 s

Obviously take equilibrium before rope snaps to solve for theta and then solve for acceleration and then solve for time but I don't get that answer if you could show me how to get that answer that would be great

To solve this problem and find the time it takes for the crate to reach a speed of 5.0 m/s, you need to follow a few steps:

Step 1: Determine if static friction is present before the rope snaps.
Since the crate is held stationary on the incline by a rope, there must be static friction acting upward to balance the downward force of gravity. The force of static friction will equal to the tension in the rope, which is 32 N in this case.

Step 2: Solve for the angle of the incline (θ).
To find the angle of the incline, you need to use the component of the weight that is parallel to the inclined plane. Since the crate is held stationary, the force of static friction must be equal to the component of the weight parallel to the incline. This can be calculated using the formula: F_staticfriction = m * g * sin(θ), where m is the mass of the crate (25 kg) and g is the acceleration due to gravity (9.8 m/s^2). Rearranging the equation, we have sin(θ) = F_staticfriction / (m * g). Substitute the given values to find sin(θ).

Step 3: Determine the force of kinetic friction.
Once the rope snaps, the static friction is no longer present. Now, the crate will experience kinetic friction as it slides down the incline. The force of kinetic friction is given by: F_kineticfriction = μ * N, where μ is the coefficient of friction (0.73) and N is the normal force. The normal force can be calculated using the formula: N = m * g * cos(θ), where cos(θ) = adjacent / hypotenuse = F_weight / (m * g). Substitute the values to find the normal force.

Step 4: Calculate the net force.
The net force acting on the crate as it moves down the incline is given by F_net = F_weight * sin(θ) - F_kineticfriction.

Step 5: Find the acceleration of the crate.
Using Newton's second law (F_net = m * a), you can solve for the acceleration (a) of the crate. Rearranging the equation, we have a = F_net / m.

Step 6: Apply the kinematic equation to find time.
The final step is to find the time it takes for the crate to reach a speed of 5.0 m/s. You can do this using the kinematic equation: v = u + a * t, where v is the final velocity (5.0 m/s), u is the initial velocity (0 m/s since the crate was initially held stationary), a is the acceleration calculated in Step 5, and t is the unknown time.

Rearrange the equation to solve for time (t), and plug in the given values to obtain the answer.

By going through these steps, you should be able to calculate the answer of 3.9 seconds as stated in the back of the book.