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, we need to calculate the time it takes for the crate to reach a speed of 5.0 m/s after the rope snaps. Let's break down the solution into steps:

Step 1: Calculate the angle of the incline
Before the rope snaps, the tension force in the rope is balancing the component of the weight acting down the slope. This can be calculated using the equation: Tension = mg * sin(θ), where m is the mass of the crate (25 kg), g is the acceleration due to gravity (9.8 m/s^2), and θ is the angle of the incline. Rearranging the equation, we get: sin(θ) = Tension / (m * g). Plugging in the given values, we have: sin(θ) = 32 N / (25 kg * 9.8 m/s^2). Solve for θ by taking the inverse sine (sin^-1) of both sides.

Step 2: Calculate the acceleration
After the rope snaps, the only force acting on the crate in the downward direction is the component of the weight parallel to the incline. This force can be calculated using the equation: Force parallel = mg * sin(θ). The friction force acting up the incline can be calculated using the equation: Friction force = μ * mg * cos(θ), where μ is the coefficient of friction (0.73) and cos(θ) is the component of gravity perpendicular to the incline. Subtracting the friction force from the force parallel, we get the net force: Net force = Force parallel - Friction force. The acceleration can be calculated using Newton's second law: a = Net force / m.

Step 3: Calculate the time
To calculate the time it takes for the crate to reach a speed of 5.0 m/s, we can use the equation: Δv = a * t, where Δv is the change in velocity (5.0 m/s in this case), a is the acceleration, and t is the time. Rearranging the equation, we have: t = Δv / a. Plugging in the given values, we get: t = 5.0 m/s / acceleration.

By following these steps, you should be able to calculate the time it takes for the crate to reach a speed of 5.0 m/s.