To move a large crate across a rough floor, you push on it with a force F at an angle of 21¨¬ below the horizontal. The mass of the crate is 31 kg, ¥ìs is 0.57, and ¥ìk is 0.45. Find the force necessary to start the crate moving
x: Fcosα –F(fr) =0
y: N-mg-Fsinα = 0 =>
N= mg+Fsinα
Fcosα =F(fr) =μ⒦•N = μ⒦• (mg+Fsinα)
F= μ⒦•m•g/ (cosα - μ⒦•sinα) =
=0.45•31•9.8/(cos40-0.45•sin40) = 286.7 N
(this is the force that necessary to move)
F= μ⒮•m•g/ (cosα - μ⒮•sinα) =
=0.57•31•9.8/(cos40-0.57•sin40) = 433.3 N
(this is the force that necessary to start the motion)
To find the force necessary to start the crate moving, we need to consider the forces acting on the crate and apply Newton's second law of motion.
1. Identify the forces acting on the crate:
- The force you apply by pushing the crate (F).
- The gravitational force pulling the crate downward (weight, mg), where m is the mass of the crate and g is the acceleration due to gravity (9.8 m/s^2).
2. Determine the horizontal and vertical components of the forces:
- The horizontal component of the force you apply is Fx = F*cos(21°).
- The vertical component of the force you apply is Fy = F*sin(21°).
- The vertical component of the weight is W = mg.
3. Apply Newton's second law of motion in the horizontal direction:
Since the crate is initially at rest, the net force acting on it in the horizontal direction is the force of static friction (Fs) opposing the applied force. Once the force of static friction is overcome, the crate will start moving.
In the horizontal direction: Fx - Fs = 0
4. Determine the force of static friction:
The force of static friction can be calculated using the equation: Fs = μs * N,
where μs is the coefficient of static friction and N is the normal force.
- The normal force (N) is equal to the weight of the crate (mg), since the crate is on a rough horizontal floor.
- The coefficient of static friction (μs) is provided as 0.57.
So, Fs = μs * N = μs * mg
5. Substitute the values into the equation to solve for Fs:
Fs = 0.57 * (31 kg * 9.8 m/s^2)
6. Rearrange the equation to solve for F:
Fx - Fs = 0
Fx = Fs
F*cos(21°) = 0.57 * (31 kg * 9.8 m/s^2)
F = (0.57 * 31 kg * 9.8 m/s^2) / cos(21°)
Now, calculate F using the given values.