A 36-N force, parallel to the incline, is required to push a certain crate at a velocity of 9.1 up a frictionless incline that is 45° above the horizontal. The mass of the crate is:

Fap = mg*sin45 = 36 N.

m = 36/g*sin45 = 5.2 kg

Fap = Force applied.

To solve this problem, we can use the equation relating force, mass, and acceleration:

Force = mass * acceleration

In this case, the force pushing the crate up the incline is given as 36 N. The angle of the incline is 45°. Since the incline is frictionless, the acceleration can be calculated using the formula:

acceleration = (gravity * sin(angle))

where gravity is the acceleration due to gravity, which is approximately 9.8 m/s^2.

Given that the acceleration is related to the force by:

acceleration = force / mass

we can rearrange this equation to solve for mass:

mass = force / acceleration

Substituting the given values:

force = 36 N
acceleration = gravity * sin(angle)

angle = 45°
gravity = 9.8 m/s^2

acceleration = 9.8 * sin(45°)

mass = 36 / (9.8 * sin(45°))

Now we can calculate the mass:

mass ≈ 36 / (9.8 * 0.7071)

mass ≈ 5.05 kg

Therefore, the mass of the crate is approximately 5.05 kg.

To find the mass of the crate, we can use the formula:

Force = mass * acceleration (F = m * a)

In this case, the force required to push the crate up the incline is given as 36 N. However, we need to determine the acceleration of the crate first.

The acceleration of the crate can be found using the equation:

acceleration = gravitational force component along the incline / mass

The gravitational force component along the incline can be calculated using the formula:

gravitational force component along the incline = mass * gravitational acceleration * sin(angle of incline)

In this case, the angle of incline is given as 45° and the gravitational acceleration is approximately 9.8 m/s^2.

So, the gravitational force component along the incline is:

gravitational force component along the incline = mass * 9.8 * sin(45°)

Now, we can plug this into the equation for acceleration:

acceleration = (mass * 9.8 * sin(45°)) / mass

Since the mass of the crate appears on both sides of the equation, we can cancel it out:

acceleration = 9.8 * sin(45°)

Now that we have the acceleration, we can use the force and acceleration to find the mass of the crate:

36 N = mass * (9.8 * sin(45°))

Solving for mass:

mass = 36 N / (9.8 * sin(45°))

Calculating this value will give us the mass of the crate.