To move a large crate across a rough floor, you push on it with a force (F) at an angle of 21 degrees below the horizontal. Find the force necessary to start the crate moving, given the mass of the crate is 32 kg and the coefficient of static friction between the crate and the floor is 0.57.
I know that the formulas used are F=ma and I can't find acceleration. I also drew a free body diagram and tried to use trig. to figure out the problem and our teacher gave us the answer sheet we just have to show how to get to the answer (the answer to this problem is 250 N)
To find the force necessary to start the crate moving, you need to consider the forces acting on the crate and use the concept of static friction.
Let's identify the forces at play:
1. Weight (W): The force due to the weight of the crate, acting vertically downwards. It can be calculated as W = m * g, where m is the mass of the crate (32 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
2. Normal force (N): The force exerted by the floor on the crate, perpendicular to the floor. It is equal in magnitude, but opposite in direction, to the weight of the crate, so N = -W.
3. Applied force (F): The force you are pushing on the crate with, at an angle of 21 degrees below the horizontal.
4. Frictional force (f): The force of static friction between the crate and the floor, opposing the applied force.
For the crate to start moving, the applied force (F) must overcome the maximum static friction (fs_max). The formula for static friction is fs_max = μs * N, where μs is the coefficient of static friction (0.57) and N is the normal force.
In our case, the normal force (N) is equal to the weight of the crate, so N = -W = -m * g.
Now, let's calculate the value of the normal force:
N = -m * g
N = -(32 kg) * (9.8 m/s^2)
N = -313.6 N
Substituting N into the formula for static friction:
fs_max = μs * N
fs_max = (0.57) * (-313.6 N)
fs_max = -178.952 N
Since friction always opposes motion, the maximum static friction is equal in magnitude to the applied force, but in the opposite direction: fs_max = -F.
Therefore, we rearrange the equation to solve for the applied force:
-F = fs_max
F = -fs_max
F = -(-178.952 N)
F = 178.952 N
The force necessary to start the crate moving is 178.952 N. However, note that the negative sign indicates that the applied force is in the opposite direction of the force of static friction.
Thus, to get the magnitude of the force required to start the crate moving, you simply remove the negative sign:
|F| = 178.952 N
Therefore, the force necessary to start the crate moving is approximately 179 N.
To find the force necessary to start the crate moving, we need to consider two forces: the force you apply to the crate (F) and the force of friction (f) between the crate and the floor.
Let's break down the problem step by step.
1. The force you apply to the crate (F) can be divided into two components: horizontal and vertical. The vertical component does not affect the crate's motion on the horizontal surface, so we'll focus on the horizontal component.
2. The horizontal component of the force you apply is given by F_horizontal = F * cos(21°). This component is responsible for overcoming the force of friction and moving the crate.
3. The force of friction (f) can be calculated using the formula f = μ * N, where μ is the coefficient of static friction and N is the normal force between the crate and the floor. The normal force (N) is equal to the weight of the crate, which is N = m * g, where m is the mass of the crate and g is the acceleration due to gravity (approximately 9.8 m/s^2).
4. Since the crate is initially at rest, the force of static friction (fs) counteracts the applied force (F_horizontal). Therefore, fs = F_horizontal.
Now we can substitute the given values and solve for the force necessary to start the crate moving.
F = fs = μ * N = μ * m * g = 0.57 * 32 kg * 9.8 m/s^2 = 178.336 N
F_horizontal = F * cos(21°) = 178.336 N * cos(21°) = 168.67 N
Therefore, the force necessary to start the crate moving is approximately 168.67 N.
Note: It's important to ensure that the force applied (F) is greater than or equal to the force necessary to start the crate moving (F_horizontal). In this case, since the force applied is known to be 250 N, it exceeds the required force, so the crate would start moving.