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Chapter 4: Problem 41

A crate of oranges slides down an inclined plane without friction. If it is released from rest and reaches a speed of \(5.832 \mathrm{~m} / \mathrm{s}\) after sliding a distance of \(2.29 \mathrm{~m},\) what is the angle of inclination of the plane with respect to the horizontal?

Short Answer

Expert verified
Answer: The angle of inclination of the plane is approximately \(48.7^{\circ}\).

Step by step solution

01

List the given information

Initial velocity \((v_i) = 0 \mathrm{~m/s}\) Final velocity \((v_f) = 5.832 \mathrm{~m/s}\) Distance \((d) = 2.29 \mathrm{~m}\)
02

Find the acceleration

To find the acceleration, we can use the kinematic equation: $$ v_f^2 = v_i^2 + 2ad $$ Since initial velocity \((v_i)\) is 0, the equation is reduced to: $$ v_f^2 = 2ad $$ Rearrange the equation to solve for acceleration: $$ a = \frac{v_f^2}{2d} $$ Plug in the values: $$ a = \frac{(5.832)^2}{2(2.29)} $$ and calculate the acceleration: $$ a \approx 7.432 \mathrm{~m/s^2} $$
03

Find the angle of inclination

Now, we will use the acceleration value we found in the previous step to find the angle of inclination. We know that on an inclined plane, the component of gravitational acceleration \(g\) along the plane is given by: $$ a = g \sin{\theta} $$ where \(\theta\) is the angle of inclination of the plane. We can rearrange the equation to solve for the angle of inclination: $$ \theta = \sin^{-1}(\frac{a}{g}) $$ Plug in the values: $$ \theta = \sin^{-1}(\frac{7.432}{9.81}) $$ and calculate the angle of inclination: $$ \theta \approx 48.7^{\circ} $$ The angle of inclination of the plane with respect to the horizontal is approximately \(48.7^{\circ}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inclined Plane
An inclined plane is a flat surface tilted at an angle relative to the horizontal ground. It's used to move objects up or down with less effort compared to lifting them directly. In our exercise, we deal with an inclined plane free of friction, which means there's no resistance affecting the motion of the sliding crate.

When objects are placed on an inclined plane, they naturally tend to move downward due to gravity. However, the presence of the inclined plane changes how gravity affects the object. Instead of moving objects straight down, gravity is split into two components: one parallel and one perpendicular to the surface of the plane.

Understanding inclined planes helps in solving problems related to simple machines, ramps, and even roller coasters. Beyond physics, you experience inclined planes in your everyday life, such as in ramps for wheelchair access.
Acceleration
Acceleration describes how quickly an object's velocity is changing over time. In this exercise, the crate moved from rest, which means its initial velocity was zero, and then reached a final velocity as it slid down the inclined plane.

The acceleration can be calculated using the kinematic equation:
  • Initial velocity (\(v_i\)) was 0 m/s
  • Final velocity (\(v_f\)) was 5.832 m/s
  • Distance (\(d\)) was 2.29 m
Using the formula \(v_f^2 = v_i^2 + 2ad\), you can find the acceleration \(a\). For this scenario:
  • \(a = \frac{v_f^2}{2d} \approx 7.432 \mathrm{~m/s^2}\)
Acceleration is essential in understanding how forces cause changes in motion and is crucial for predicting future motion patterns of objects.
Gravitational Force
Gravitational force is the force of attraction between two masses. On Earth, this force pulls objects towards the center of the planet, giving them weight. This force can be broken down into components when an object is on an inclined plane.

The equation \(a = g \sin{\theta}\) illustrates how gravitational acceleration affects the object on an inclined plane. Here:
  • \(g\) is the acceleration due to gravity, approximately \(9.81 \mathrm{~m/s^2}\)
  • \(\sin{\theta}\) represents the component of gravity parallel to the inclined plane
This component is responsible for pulling the crate down the plane, causing it to accelerate. The other component, perpendicular to the plane, is matched by the normal force and does not contribute to the motion. Understanding gravitational force in terms of its components is vital for analyzing motion on slopes and ramps.
Trigonometry
Trigonometry helps us relate the angles and sides of triangles to solve various physics problems. In the context of our inclined plane scenario, trigonometry helps to connect the angle of inclination with the motion of the object.

To find the angle of the inclined plane, we use the sine function, which links our calculated acceleration and the gravitational force:
  • \(a = g \sin{\theta}\)
  • \(\theta = \sin^{-1}(\frac{a}{g})\)
Using trigonometric functions allows one to calculate the angle of inclination. In the given example, the angle \(\theta\) came out to be approximately 48.7 degrees.

Trigonometry is an essential tool in physics, aiding in a variety of calculations, such as determining inclines, forces, and even rotational dynamics.

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Most popular questions from this chapter

A pinata of mass \(M=12\) kg hangs on a rope of negligible mass that is strung between the tops of two vertical poles. The horizontal distance between the poles is \(D=2.0 \mathrm{~m}\), the top of the right pole is a vertical distance \(h=0.50 \mathrm{~m}\) higher than the top of the left pole, and the total length of the rope between the poles is \(L=3.0 \mathrm{~m}\). The pinata is attached to a ring, with the rope passing through the center of the ring. The ring is frictionless, so that it can slide freely

4.40 A store sign of mass \(4.25 \mathrm{~kg}\) is hung by two wires that each make an angle of \(\theta=42.4^{\circ}\) with the ceiling. What is the tension in each wire?

A bosun's chair is a device used by a boatswain to lift himself to the top of the mainsail of a ship. A simplified device consists of a chair, a rope of negligible mass, and a frictionless pulley attached to the top of the mainsail. The rope goes over the pulley, with one end attached to the chair, and the boatswain pulls on the other end, lifting himself upward. The chair and boatswain have a total mass \(M=90.0 \mathrm{~kg}\). a) If the boatswain is pulling himself up at a constant speed, with what magnitude of force must he pull on the rope? b) If, instead, the boatswain moves in a jerky fashion, accelerating upward with a maximum acceleration of magnitude \(a=2.0 \mathrm{~m} / \mathrm{s}^{2},\) with what maximum magnitude of force must he null on the rone?

\(\bullet 4.60\) A block of mass \(m_{1}=21.9 \mathrm{~kg}\) is at rest on a plane inclined at \(\theta=30.0^{\circ}\) above the horizontal. The block is connected via a rope and massless pulley system to another block of mass \(m_{2}=25.1 \mathrm{~kg}\), as shown in the figure. The coefficients of static and kinetic friction between block 1 and the inclined plane are \(\mu_{s}=0.109\) and \(\mu_{k}=0.086\) respectively. If the blocks are released from rest, what is the displacement of block 2 in the vertical direction after 1.51 s? Use positive numbers for the upward direction and negative numbers for the downward direction.

A spring of negligible mass is attached to the ceiling of an elevator. When the elevator is stopped at the first floor, a mass \(M\) is attached to the spring, stretching the spring a distance \(D\) until the mass is in equilibrium. As the elevator starts upward toward the second floor, the spring stretches an additional distance \(D / 4\). What is the magnitude of the acceleration of the elevator? Assume the force provided by the spring is linearly proportional to the distance stretched by the spring.
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