Sure, let's solve the problem step-by-step.
1. Identify the given values:
- Mass of the crate ([tex]\( m \)[/tex]): 75.5 kg
- Coefficient of kinetic friction ([tex]\( \mu_k \)[/tex]): 0.262
- Acceleration due to gravity ([tex]\( g \)[/tex]): 9.81 m/s²
2. Calculate the normal force ([tex]\( N \)[/tex]):
The normal force is the force exerted by a surface to support the weight of an object resting on it. On flat ground, this equals the weight of the object.
[tex]\[
N = m \times g
\][/tex]
Substituting the given values:
[tex]\[
N = 75.5\, \text{kg} \times 9.81\, \text{m/s}^2 = 740.655\, \text{N}
\][/tex]
3. Calculate the frictional force ([tex]\( F_{\text{friction}} \)[/tex]):
The frictional force can be calculated using the coefficient of kinetic friction and the normal force.
[tex]\[
F_{\text{friction}} = \mu_k \times N
\][/tex]
Substituting the known values:
[tex]\[
F_{\text{friction}} = 0.262 \times 740.655\, \text{N} = 194.05161\, \text{N}
\][/tex]
4. Determine the acceleration ([tex]\( a \)[/tex]):
According to Newton's second law, force is equal to mass times acceleration ([tex]\( F = m \times a \)[/tex]). The frictional force is what causes the crate to decelerate.
[tex]\[
F_{\text{friction}} = m \times a
\][/tex]
Rearrange to solve for acceleration ([tex]\( a \)[/tex]):
[tex]\[
a = \frac{F_{\text{friction}}}{m}
\][/tex]
Substituting the values:
[tex]\[
a = \frac{194.05161\, \text{N}}{75.5\, \text{kg}} = 2.57022\, \text{m/s}^2
\][/tex]
Since the frictional force is acting in the direction opposite to the motion of the crate, the acceleration is negative (indicating deceleration):
[tex]\[
a = -2.57022\, \text{m/s}^2
\][/tex]
Therefore, the acceleration of the crate is [tex]\( -2.57022 \, \text{m/s}^2 \)[/tex].
