The weight of a book on an inclined plane can be resolved into two vector components, one along the plane, the other perpendicular to it.
At what incline angle are the components equal? I was kind of thinking 45 degress??????
At what incline angle is the parallel component equal to zero? I was thinking 90 degrees?????
At what incline angle is the parallel component equal to the weight?
I have no idea???
HELP!
(a) At what incline angle are the components equal?
(_) 0°
(_) 30°
(o) 45°
(_) 60°
(_) 90°
correct check mark
(b) At what incline angle is the parallel component equal to zero?
(o) 0°
(_) 30°
(_) 45°
(_) 60°
(_) 90°
correct check mark
(c) At what incline angle is the parallel component equal to the weight?
(_) 0°
(_) 30°
(_) 45°
(_) 60°
(o) 90°
correct check mark
Well, aren't you just inclined to ask some interesting questions? Let's see if I can shed some light-heartedness on the subject:
First up, when the weight of the book on the inclined plane is resolved into two vector components – one along the plane and the other perpendicular to it – the angles you're asking about come into play.
When the components are equal, you're right on the money with 45 degrees! So congratulations, you've got a good sense of balance there.
Now, when it comes to the angle at which the parallel component is equal to zero, you're almost there but not quite. It's actually at 90 degrees – when the plane is perfectly vertical. So close, yet so far!
Lastly, when is the parallel component equal to the weight? Drumroll, please... It's at a whopping 0 degrees, my friend! That's when the inclined plane is completely horizontal.
I hope I've been able to turn your frown upside down with these little tidbits of information. Remember, learning can be a clown-tastic adventure!
To solve these questions, we need to understand the components of weight on an inclined plane.
1. At what incline angle are the components equal?
To find the incline angle at which the components are equal, we can set up a right triangle. Let's call the weight of the book "W". The weight can be resolved into two components: one along the plane (called the "parallel component") and the other perpendicular to it (called the "perpendicular component").
In a right triangle, the weight W is the hypotenuse, the parallel component is the adjacent side, and the perpendicular component is the opposite side.
By using trigonometric functions, we can relate the angle of inclination (θ) to the parallel and perpendicular components:
tan(θ) = perpendicular component / parallel component
Since we want the parallel and perpendicular components to be equal, we can set up the equation as:
tan(θ) = 1
To find the angle, we take the inverse tangent (arctan) of both sides:
θ = arctan(1)
The value of arctan(1) is approximately 45 degrees.
So, the incline angle at which the components are equal is 45 degrees.
2. At what incline angle is the parallel component equal to zero?
The parallel component of weight is responsible for the book's motion along the inclined plane. When the parallel component is zero, the book will not move along the plane but will only exert a force perpendicular to it.
In this case, the angle at which the parallel component is zero is when the plane is vertical, or 90 degrees.
3. At what incline angle is the parallel component equal to the weight?
When the parallel component is equal to the weight, the book is on a steep incline where the force parallel to the plane counteracts the weight of the book.
To calculate this angle, we can set up the equation:
sin(θ) = parallel component / weight
Since we want the parallel component to be equal to the weight, we can rewrite the equation as:
sin(θ) = 1
Taking the inverse sine (arcsin) of both sides, we find:
θ = arcsin(1)
The value of arcsin(1) is 90 degrees.
Therefore, the incline angle at which the parallel component is equal to the weight is 90 degrees.
I hope this helps! Let me know if you have any further questions.
To find the answers to your questions, let's break down the problem step by step.
Firstly, when a book is on an inclined plane, its weight can be resolved into two components: one parallel to the plane (the parallel component) and one perpendicular to the plane (the perpendicular component).
1. At what incline angle are the components equal?
To determine the incline angle at which the components are equal, we can use trigonometry. Let's call the weight of the book "W" and the angle of inclination "θ".
The perpendicular component of the weight is given by W * sin(θ), and the parallel component is given by W * cos(θ).
For the components to be equal, we need to set them equal to each other and solve for θ:
W * sin(θ) = W * cos(θ)
Divide both sides by W:
sin(θ) = cos(θ)
Now, we can find the angle θ at which this equation holds true. Taking the inverse sine of both sides:
θ = arcsin(cos(θ))
Using a scientific calculator or any trigonometric identity, we find that θ = 45 degrees. So, you were right!
2. At what incline angle is the parallel component equal to zero?
The parallel component of the weight becomes zero when the book is on a completely vertical inclination. In other words, when the inclined plane is perpendicular to the ground, the angle of inclination is 90 degrees. Therefore, you were right again!
3. At what incline angle is the parallel component equal to the weight?
To find the angle at which the parallel component is equal to the weight (W), we need to set up the equation:
W * cos(θ) = W
Dividing both sides by W:
cos(θ) = 1
Taking the inverse cosine of both sides:
θ = arccos(1)
The arccosine of 1 is 0, so the angle θ is 0 degrees. This means that the incline angle would be 0, or the book would be placed flat on the surface. Therefore, the parallel component is equal to the weight when the angle of inclination is 0 degrees.
I hope this explanation helps! Let me know if you have any more questions.