You push a large 32 kg crate across the floor with a force F=330 N at an angle of 21 degrees below the horizontal. If the coefficient of kinetic friction between the crate and the floor is 0.45, what is the acceleration of the crate?

To find the acceleration of the crate, we need to consider the forces acting on it. In this case, there are two main forces: the horizontal component of the applied force and the force of kinetic friction.

First, let's find the horizontal component of the applied force. We can calculate it using the formula:

F_horizontal = F * cos(angle)

where F is the applied force and angle is the angle below the horizontal. Plugging in the values, we have:

F_horizontal = 330 N * cos(21 degrees)

Next, let's find the force of kinetic friction. We can calculate it using the formula:

Frictional force = coefficient of kinetic friction * normal force

The normal force is the force exerted by a surface to support the weight of an object resting on it. In this case, the normal force is equal to the weight of the crate, which can be calculated as:

Weight = mass * gravitational acceleration

where mass is the mass of the crate and gravitational acceleration is approximately 9.8 m/s^2. Plugging in the values, we have:

Weight = 32 kg * 9.8 m/s^2

Now we can find the force of kinetic friction:

Frictional force = 0.45 * Weight

Finally, to find the acceleration, we need to use Newton's second law of motion, which states:

Force = mass * acceleration

In this case, the net force is the difference between the horizontal component of the applied force and the force of kinetic friction:

Net force = F_horizontal - Frictional force

Plugging in the values, we have:

Net force = F_horizontal - (0.45 * Weight)

Now we can rearrange the equation to solve for acceleration:

acceleration = Net force / mass

Plugging in the values and using the equation above, we can now calculate the acceleration of the crate.

To find the acceleration of the crate, we need to consider the forces acting on it.

The force applied to the crate, F, can be broken down into horizontal and vertical components. The horizontal component of the force, Fx, is given by:

Fx = F * cosθ,

where θ is the angle below the horizontal.

Substituting the given values:

Fx = 330 N * cos21°.

Using a calculator:

Fx ≈ 311.47 N.

The frictional force, Ff, between the crate and the floor can be calculated using the equation:

Ff = μ * N,

where μ is the coefficient of kinetic friction and N is the normal force.

The normal force, N, is equal to the weight of the crate, which can be calculated as:

N = m * g,

where m is the mass of the crate and g is the acceleration due to gravity (9.8 m/s^2).

Substituting the given values:

N = 32 kg * 9.8 m/s^2,

N ≈ 313.6 N.

Now, substituting these values into the equation for the frictional force:

Ff = 0.45 * 313.6 N,

Ff ≈ 141.12 N.

Since the crate is moving, the frictional force is equal to the product of the mass of the crate and the acceleration, Ff = m * a.

Rearranging this equation to solve for the acceleration:

a = Ff / m,

a = 141.12 N / 32 kg,

a ≈ 4.41 m/s^2.

Therefore, the acceleration of the crate is approximately 4.41 m/s^2.