Question

Mass on a plane \(\mathrm{A}\) 100-kg object rests on an inclined plane at an angle of \(30^{\circ}\) to the floor. Find the components of the force perpendicular to and parallel to the plane. (The vertical component of the force exerted by an object of mass \(m\) is its weight, which is \(m g\) where \(g=9.8 \mathrm{m} / \mathrm{s}^{2}\) is the acceleration due to gravity.)

Short answer

The components of the force perpendicular to and parallel to the inclined plane are approximately 848.28 N and 490 N, respectively.

Step-by-step solution

Calculate the Weight of the Object

To find the vertical component of the force exerted by the object, we need to calculate its weight. The weight of the object can be found using the formula: Weight = mass × gravitational acceleration. Given, mass (m) = 100 kg gravitational acceleration (g) = 9.8 m/s² Weight = m × g = 100 kg × 9.8 m/s² = 980 N (Newtons)

Find the Components of the Force

Now, we need to find the components of the force, which are perpendicular to and parallel to the inclined plane. To do that, we can use the following trigonometric relationships: 1. Perpendicular component of the force (F⊥) = Weight × cos(θ) 2. Parallel component of the force (F∥) = Weight × sin(θ) Given, θ = 30°

Calculate the Perpendicular Component of the Force

To find the perpendicular component of the force, we can apply the first trigonometric relationship: F⊥ = Weight × cos(θ) = 980 N × cos(30°) Use a calculator to find the cosine of 30°: cos(30°) ≈ 0.866 F⊥ ≈ 980 N × 0.866 ≈ 848.28 N The perpendicular component of the force is approximately 848.28 N.

Calculate the Parallel Component of the Force

To find the parallel component of the force, we can apply the second trigonometric relationship: F∥ = Weight × sin(θ) = 980 N × sin(30°) Use a calculator to find the sine of 30°: sin(30°) ≈ 0.5 F∥ ≈ 980 N × 0.5 ≈ 490 N The parallel component of the force is approximately 490 N. To summarize, the components of the force perpendicular to and parallel to the inclined plane are approximately 848.28 N and 490 N, respectively.

Key concepts

Force Components

When an object is placed on an inclined plane, its weight force can be split into two parts: perpendicular to the plane and parallel to the plane. Understanding these components is crucial to solving problems related to inclined planes.

• The perpendicular component is responsible for the object's normal force, which is the force pressing it into the surface.
• The parallel component causes the object to slide down the plane if unopposed by friction or other forces.

Breaking down the weight force into these components helps us analyze the object's behavior. This is particularly important in physics and engineering, where analyzing how forces act in different directions allows us to make precise calculations.

Trigonometric Relationships

To accurately find the force components on an inclined plane, we use trigonometric relationships. These relationships are central to decomposing the force of gravity acting on the object. For an incline at angle

• The perpendicular component is calculated using the cosine function: \[F_\perp = \text{Weight} \times \cos(\theta)\]
• The parallel component uses the sine function: \[F_{\parallel} = \text{Weight} \times \sin(\theta)\]

Here, \(\theta\) is the angle of the inclined plane, and using these functions allows us to separate the weight into its respective components accurately. These calculations help predict how the object will move or remain stationary on the incline.

Gravity Force

Gravity is the force that pulls objects downward towards Earth. On an inclined plane, gravity acts on the object to create its weight force, which can be calculated by multiplying its mass by the gravitational acceleration (\(g = 9.8 \, \text{m/s}^2\)). Gravity always acts vertically downward, but when dealing with an inclined plane, we need to consider how this vertical force splits into two components due to the angle of the incline.

• The weight of the object on the inclined plane is given by: \[\text{Weight} = m \times g = 100 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 980 \, \text{N}\]

This weight is then used to find the force components, enabling us to understand how gravity influences the object's behavior on the slope. These calculations are essential in physics to predict movement and ensure stability in practical situations.