A box is sliding up an incline that makes an angle of 12.0° with respect to the horizontal. The coefficient of kinetic friction between the box and the surface of the incline is 0.180. The initial speed of the box at the bottom of the incline is 1.30 m/s. How far does the box travel along the incline before coming to rest?
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To find out how far the box travels along the incline before coming to rest, we need to calculate the distance using the given information. Here's how you can solve it step by step:
First, let's analyze the forces acting on the box:
1. Gravitational force (Fg): The downward force acting on the box due to its weight. The magnitude of this force can be calculated as Fg = m * g, where m is the mass of the box and g is the acceleration due to gravity (approximately 9.8 m/s²).
2. Normal force (Fn): The perpendicular force exerted by the incline on the box. This is equal in magnitude but opposite in direction to the vertical component of the gravitational force, which is Fn = m * g * cos(θ), where θ is the angle of the incline.
3. Frictional force (Ff): The force opposing the motion of the box. The magnitude of the frictional force can be calculated as Ff = μ * Fn, where μ is the coefficient of kinetic friction.
4. Component of gravitational force along the incline (Fg_parallel): This force component acts parallel to the incline and contributes to the acceleration of the box. The magnitude of Fg_parallel can be calculated as Fg_parallel = m * g * sin(θ).
With these forces identified, we can apply Newton's second law to determine the acceleration of the box:
Net force = m * a,
where m is the mass of the box and a is the acceleration of the box.
Since the box is coming to rest, the net force is equal to zero. Hence, we have:
Fg_parallel - Ff = 0.
(m * g * sin(θ)) - (μ * Fn) = 0.
Now, let's substitute the known values into the equation:
m * g * sin(θ) - μ * m * g * cos(θ) = 0.
Next, we can simplify the equation by canceling out the mass "m" and the gravitational acceleration "g":
sin(θ) - μ * cos(θ) = 0.
Let's solve this equation to find the value of the angle θ:
sin(θ) = μ * cos(θ).
tan(θ) = μ.
θ = arctan(μ).
Substituting the given coefficient of kinetic friction in the equation, we find:
θ = arctan(0.180).
Using a scientific calculator or trigonometric table, we can find the angle θ to be approximately 9.825°.
Now that we have determined the angle of the incline, we can calculate the distance traveled along the incline before the box comes to a rest. The equation for the distance traveled along an incline can be given as follows:
distance = (initial velocity²) / (2 * acceleration),
where the initial velocity is 1.30 m/s and the acceleration is g * sin(θ).
Substituting the values into the equation, we get:
distance = (1.30²) / (2 * 9.8 * sin(9.825°)).
Calculating this will give you the final answer for the distance traveled by the box along the incline before coming to rest.