To move a large crate across a rough floor, you push on it with a force at an angle of 21 below the horizontal, as shown in the figure.
Find the acceleration of the crate if the applied force is 400 , the mass of the crate is 32 and the coefficient of kinetic friction is 0.31.
Fc = mg = 32kg * 9.8Nkg = 313.6N = Force of crate.
Fp = mg*sin(0) = 313.6sin(0) = 0 N =
Force parallel to the plane = Fh.
Fv = mg*cos(0) + Fap*sin21,
Fv = 313.6cos(0) + 400sin21 = 456.9N =
Force perpendicular to plane. Fap = Applied force.
Ff = u*Fv = 0.31 * 456.9 = 141.7N =
Force of friction.
Fn = Fap*cos21 - Fp - Ff,
Fn = 400cos21 - 0 - 141.7 = 231.7N =
Net force.
a = Fn / m = 231.7 / 32 = 7.2m/s^2.
To determine the acceleration of the crate, we need to calculate the net force acting on it. The force can be broken down into horizontal and vertical components.
The horizontal component of the force can be found using the formula:
F_horizontal = F_applied * cos(theta),
where F_applied is the applied force and theta is the angle of the force below the horizontal.
F_horizontal = 400 * cos(21) ≈ 374.6 N.
The vertical component of the force can be found using the formula:
F_vertical = F_applied * sin(theta),
where F_applied is the applied force and theta is the angle of the force below the horizontal.
F_vertical = 400 * sin(21) ≈ 142.1 N.
Next, we need to find the force of friction acting on the crate. The force of friction can be calculated using the formula:
F_friction = coefficient_of_friction * normal_force,
where coefficient_of_friction is the coefficient of kinetic friction and normal_force is the force perpendicular to the surface.
The normal force can be calculated using the formula:
normal_force = mass * gravitational_acceleration,
where mass is the mass of the crate and gravitational_acceleration is the acceleration due to gravity.
normal_force = 32 kg * 9.8 m/s^2 = 313.6 N.
F_friction = 0.31 * 313.6 N ≈ 97.2 N.
Now, we can calculate the net force acting on the crate in the horizontal direction:
net_force_horizontal = F_horizontal - F_friction = 374.6 N - 97.2 N = 277.4 N.
Finally, we can calculate the acceleration of the crate using Newton's second law:
net_force_horizontal = mass * acceleration,
where mass is the mass of the crate and acceleration is the acceleration of the crate.
277.4 N = 32 kg * acceleration.
Solving for acceleration, we find:
acceleration ≈ 8.67 m/s^2.
Therefore, the acceleration of the crate is approximately 8.67 m/s^2.
To find the acceleration of the crate, we need to consider three forces acting on it: the applied force, the force of friction, and the force of gravity.
1. Applied force: The applied force is 400 N, and it is applied at an angle of 21 degrees below the horizontal. We need to find the horizontal and vertical components of this force.
Horizontal component: F_horizontal = F_applied * cos(theta)
F_horizontal = 400 N * cos(21 degrees)
F_horizontal ≈ 377.33 N
Vertical component: F_vertical = F_applied * sin(theta)
F_vertical = 400 N * sin(21 degrees)
F_vertical ≈ 142.14 N
2. Force of friction: The force of friction between two surfaces can be calculated using the formula F_friction = μ * N, where μ is the coefficient of kinetic friction and N is the normal force. In this case, the normal force is equal to the weight of the crate.
Normal force: N = mass * gravity
N = 32 kg * 9.8 m/s²
N ≈ 313.6 N
Force of friction: F_friction = coefficient of kinetic friction * normal force
F_friction = 0.31 * 313.6 N
F_friction ≈ 97.216 N
3. Force of gravity: The force of gravity is the weight of the crate, which can be calculated using the formula weight = mass * gravity.
Force of gravity: F_gravity = mass * gravity
F_gravity = 32 kg * 9.8 m/s²
F_gravity ≈ 313.6 N
Considering these forces, we can now find the net force acting on the crate.
Net force: F_net = F_horizontal - F_friction - F_gravity
F_net = 377.33 N - 97.216 N - 313.6 N
F_net ≈ -33.486 N
Since the applied force and force of friction are in opposite directions, we get a negative value for the net force. This indicates that the crate will experience a deceleration or a negative acceleration.
Finally, we calculate the acceleration using Newton's second law: F_net = mass * acceleration.
acceleration: acceleration = F_net / mass
acceleration = -33.486 N / 32 kg
acceleration ≈ -1.046 m/s²
Therefore, the acceleration of the crate is approximately -1.046 m/s², indicating that it is decelerating in the opposite direction of the applied force.
I assume you are pushing downward, the angle is confusing.
If you are pushing downward, some of that is pressing downward increasing fn
fn=mg+Fsin21
fhorizontal=Fcos21
horizontal force-friction= ma
F*cos21-mu*(mg+Fsin21)=m(400)
solve for F. You did notput units in the statement, you need to check them