A clerk moves a box of cans down an aisle by pulling on a strap attached to the box. The clerk pulls with a force of 188.0 N at an angle of 26° with the horizontal. The box has a mass of 30 kg, and the coefficient of kinetic friction between the box and the floor is 0.45. Find the acceleration of the box.
in m/s2 down the aisle
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Why did the box go to therapy? Because it had a lot of acceleration issues!
To calculate the acceleration of the box, we need to consider the forces acting on it. The force of gravity pulling the box down can be calculated using the formula:
Force of gravity = mass * acceleration due to gravity
Force of gravity = 30 kg * 9.8 m/s^2 (approximate value of acceleration due to gravity)
Force of gravity = 294 N
The force of friction opposing motion can be calculated using the formula:
Force of friction = coefficient of friction * normal force
Normal force = force of gravity (since the box is on a flat floor)
Normal force = 294 N
Force of friction = 0.45 * 294 N
Force of friction = 132.3 N
Now, using the angle and the force applied by the clerk, we can determine the horizontal and vertical components of the force:
Horizontal component of force = force applied * cos(angle)
Horizontal component of force = 188.0 N * cos(26°)
Vertical component of force = force applied * sin(angle)
Vertical component of force = 188.0 N * sin(26°)
The net force acting horizontally on the box is given by:
Net force = horizontal component of force - force of friction
Net force = (188.0 N * cos(26°)) - 132.3 N
Finally, we can calculate the acceleration of the box using Newton's second law:
Net force = mass * acceleration
acceleration = Net force / mass
acceleration = (188.0 N * cos(26°) - 132.3 N) / 30 kg
Solving this equation will give you the acceleration of the box down the aisle in m/s^2.
To find the acceleration of the box, we need to analyze the forces acting on it.
1. Resolve the pulling force:
The pulling force can be resolved into two components: one parallel to the floor and one perpendicular to the floor.
The component parallel to the floor (Fx) = pulling force * cos(angle)
Fx = 188.0 N * cos(26°)
The component perpendicular to the floor (Fy) = pulling force * sin(angle)
Fy = 188.0 N * sin(26°)
2. Determine the friction force:
The friction force is given by the equation Ffriction = coefficient of friction * normal force.
The normal force (Fn) is equal to the weight of the box, which is given by the equation Fn = mass * gravitational acceleration.
Fn = 30 kg * 9.8 m/s^2
The friction force (Ffriction) is then Ffriction = 0.45 * Fn
3. Calculate the net force:
The net force acting in the horizontal direction is given by Fnet = Fx - Ffriction
4. Determine the acceleration:
The acceleration of the box can be calculated using Newton's second law, which states that Fnet = mass * acceleration.
Thus, acceleration (a) = Fnet / mass
Now let's calculate the values step by step:
Calculating the component of the pulling force parallel to the floor:
Fx = 188.0 N * cos(26°)
Fx = 170.6575 N
Calculating the component of the pulling force perpendicular to the floor:
Fy = 188.0 N * sin(26°)
Fy = 78.3047 N
Calculating the normal force:
Fn = 30 kg * 9.8 m/s^2
Fn = 294 N
Calculating the friction force:
Ffriction = 0.45 * Fn
Ffriction = 0.45 * 294 N
Ffriction = 132.3 N
Calculating the net force:
Fnet = Fx - Ffriction
Fnet = 170.6575 N - 132.3 N
Fnet = 38.3575 N
Calculating the acceleration:
a = Fnet / mass
a = 38.3575 N / 30 kg
a = 1.2786 m/s^2 (rounded to four decimal places)
Therefore, the acceleration of the box down the aisle is approximately 1.2786 m/s^2.
To find the acceleration of the box, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration.
The first step is to resolve the 188.0 N force into its horizontal and vertical components. The horizontal component is given by the equation:
F_horizontal = F * cos(theta)
where F is the magnitude of the force (188.0 N) and theta is the angle (26°).
F_horizontal = 188.0 N * cos(26°)
F_horizontal = 169.8 N
The next step is to calculate the force of friction acting on the box. The force of friction can be calculated using the equation:
F_friction = coefficient_of_friction * normal_force
where coefficient_of_friction is given as 0.45 and normal_force is the force exerted by the box on the floor, which is equal to its weight.
Normal force (N) = mass * gravity
where the mass is 30 kg and gravity is approximately 9.8 m/s^2.
Normal force = 30 kg * 9.8 m/s^2
Normal force = 294 N
F_friction = 0.45 * 294 N
F_friction = 132.3 N
The net force acting on the box in the horizontal direction can be obtained by subtracting the force of friction from the horizontal component of the applied force:
Net force (F_net) = F_horizontal - F_friction
F_net = 169.8 N - 132.3 N
F_net = 37.5 N
Now, using Newton's second law, we can calculate the acceleration:
F_net = mass * acceleration
Substituting the values we have:
37.5 N = 30 kg * acceleration
Solving for acceleration:
acceleration = 37.5 N / 30 kg
acceleration ≈ 1.25 m/s^2
Therefore, the acceleration of the box is approximately 1.25 m/s^2 down the aisle.