Question

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

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

Step-by-step solution

List the given information

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

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: $$ a \approx 7.432 \mathrm{~m/s^2} $$

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 \(48.7^{\circ}\).

Key concepts

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.