A box which has a mass of 32.8 kg is placed on a plane which is inclined at 40.0°above the horizontal. The sliding friction force acting between the box and the plane is 36.4 N. What is the acceleration of the box as it slides down the plane? (Ans: 5.19m/s2) Explain why isn't acceleration negative and why isn't the Fg -x component down the slope negative as well.
You can draw your coordinate system with x pointing down the slope and y up perpendicular to the slope if you wish.
Then component of m g perpendicular to slope = m g cos 40
and component of mg down slope in x direction = m g sin 40
= 32.8 * 9.8* sin 40
= 206.6 Newtons
net force down slope = 206.6 - 36.4 = 170.2 N
a = F/m = 170.1/32.8 = 5.19 m/s^2
To find the acceleration of the box as it slides down the plane, we need to consider the force components acting on it.
The force of gravity, also known as the weight (Fg), is acting vertically downward. To determine the component of the weight acting down the slope, we need to find the force parallel to the incline.
The force parallel to the incline (Fparallel) can be calculated by multiplying the weight (Fg) by the sine of the angle of inclination (θ):
Fparallel = Fg * sin(θ)
In this case, the angle of inclination (θ) is 40.0°. The weight (Fg) can be found by multiplying the mass of the box (m) by the acceleration due to gravity (g):
Fg = m * g
Here, the mass (m) is given as 32.8 kg. The acceleration due to gravity (g) is approximately 9.8 m/s².
Substituting the values, we have:
Fg = 32.8 kg * 9.8 m/s² = 321.44 N
Fparallel = 321.44 N * sin(40.0°) = 206.29 N
Since the box is sliding down the plane, the net force acting on the box is the difference between the force parallel to the incline (Fparallel) and the sliding friction force (Ffriction):
Net force = Fparallel - Ffriction
Given that the sliding friction force (Ffriction) is 36.4 N, the net force becomes:
Net force = 206.29 N - 36.4 N = 169.89 N
Finally, we can find the acceleration (a) using Newton's second law, which states that the net force (Fnet) is equal to the mass (m) multiplied by the acceleration (a):
Fnet = m * a
Rearranging the equation, we get:
a = Fnet / m = 169.89 N / 32.8 kg ≈ 5.19 m/s²
Now, let's address your concerns about the sign of the acceleration and forces:
1. Why isn't the acceleration negative?
The acceleration in this case is positive because the box is sliding down the slope. In physics, we typically take the direction of motion as the positive direction. Since the box is moving down the incline, the acceleration is considered positive. If the box were moving up the incline, the acceleration would be negative.
2. Why isn't the Fg - x component down the slope negative?
The force parallel to the incline (Fparallel) is already accounted for in the calculations as a positive value. The negative sign is not needed because the force acting down the slope (Fparallel) is naturally positive when the box is sliding down. The negative sign would be used if the box were moving up the slope.