A 125-newton box rests on a plank 3.00 meters long, one end of which is 1.00 meters higher than the other end. Find the components of the force the box exerts
Parallel to the inclined plane=
Perpendicular to the inclined plane=
The plank is inclined
A = sin^-1 1/3 = 19.5 degrees.
Weight force parallel to plane:
125 N*sin A = 41.7 N
Perpendicular to plane:
125 N*cosA = 117.9 N
To find the components of the force the box exerts parallel and perpendicular to the inclined plane, we can apply the concept of resolving forces into components.
The weight of the box, acting vertically downwards, can be resolved into two components: one parallel to the inclined plane and the other perpendicular to the inclined plane.
Let's solve this step by step:
Step 1: Calculate the weight of the box.
Given: Weight of the box = 125 Newtons
Step 2: Calculate the component parallel to the inclined plane.
The weight of the box can be resolved into two components using trigonometry. The component parallel to the inclined plane is given by:
Force parallel = Weight of the box * sin(theta)
In this case, theta is the angle of inclination of the plank with the horizontal. Since the height difference is given (1 meter) and the length of the plank is given (3 meters), we can calculate the angle of inclination using trigonometry:
sin(theta) = opposite/hypotenuse
sin(theta) = 1/3
theta = sin^(-1)(1/3)
Using a calculator, we find this value to be approximately 19.47 degrees.
Now we can calculate the component parallel to the inclined plane:
Force parallel = 125 Newtons * sin(19.47 degrees)
Step 3: Calculate the component perpendicular to the inclined plane.
The component perpendicular to the inclined plane is given by:
Force perpendicular = Weight of the box * cos(theta)
Using the same angle of inclination obtained earlier, we can calculate the component perpendicular to the inclined plane:
Force perpendicular = 125 Newtons * cos(19.47 degrees)
Therefore, the components of the force the box exerts are:
Parallel to the inclined plane = Force parallel = 125 Newtons * sin(19.47 degrees)
Perpendicular to the inclined plane = Force perpendicular = 125 Newtons * cos(19.47 degrees)
To find the components of the force exerted by the box on the inclined plane, we need to analyze the forces acting on the box. In this case, we have the weight of the box acting vertically downward, and the normal force exerted by the inclined plane acting perpendicular to the surface.
First, let's find the weight of the box. The weight of an object is given by the formula:
Weight = mass * gravity
where mass is the mass of the object and gravity is the acceleration due to gravity. However, in this case, we are given the weight (125 Newtons), so we can directly use it.
The weight of the box is 125 Newtons.
Next, let's find the components of the weight. Since the plank is inclined, we need to analyze the forces along the plank. We can break down the weight into two components: one parallel to the inclined plane and one perpendicular to the inclined plane.
To find the component of weight parallel to the inclined plane, we need to use trigonometry. Since the plank is inclined, it forms a right-angled triangle with the vertical and horizontal axes. Let's label the angle between the inclined plane and the horizontal axis as θ.
The component of weight parallel to the inclined plane is given by the formula:
Parallel Component = Weight * sin(θ)
To find θ, we need to use the given information that one end of the plank is 1.00 meter higher than the other end. We can use this height difference and the length of the plank to calculate the sine of the angle θ. Let's call the height difference h and the length of the plank L.
In this case, h = 1.00 meters and L = 3.00 meters.
The sine of θ is given by the formula:
sin(θ) = h / L
Plugging in the values, we have:
sin(θ) = 1.00 / 3.00
Now, we can calculate the component of weight parallel to the inclined plane:
Parallel Component = 125 * sin(θ)
Finally, let's find the component of weight perpendicular to the inclined plane. This component is equal to the normal force exerted by the inclined plane, as they have equal magnitudes but opposite directions.
The component of weight perpendicular to the inclined plane (the normal force) is given by the formula:
Perpendicular Component = Weight * cos(θ)
Using the same angle θ as before (which we calculated using the height difference and length of the plank), we have:
Perpendicular Component = 125 * cos(θ)
Therefore, the components of the force exerted by the box on the inclined plane are:
Parallel to the inclined plane = 125 * sin(θ)
Perpendicular to the inclined plane = 125 * cos(θ)
You can calculate the numerical values by substituting the value of θ obtained from sin(θ) = h / L into the formulas for the parallel and perpendicular components of the force.