Question

A wedge of mass \(m=36.1 \mathrm{~kg}\) is located on a plane that is inclined by an angle \(\theta=21.3^{\circ}\) with respect to the horizontal. A force \(F=302.3 \mathrm{~N}\) in the horizontal direction pushes on the wedge, as shown in the figure. The coefficient of kinetic friction between the wedge and the plane is 0.159 What is the acceleration of the wedge along the plane?

Short answer

Answer: The acceleration of the wedge along the inclined plane is approximately 2.95 m/s².

Step-by-step solution

Diagram and Forces

Draw a diagram of the wedge on the inclined plane, with the horizontal force F acting to the right. Label the angle of inclination (θ), and indicate the weight of the wedge (mg) acting vertically downward.

Find the weight components

The weight of the wedge can be broken down into components parallel and perpendicular to the inclined plane: - Parallel component: \(mg \sin \theta\) - Perpendicular component: \(mg \cos \theta\)

Calculate the friction force

The friction force acting on the wedge is given by: - \(F_f = \mu F_n\) Where F_f is the friction force and F_n is the normal force. In this case, the normal force is equal to the perpendicular component of the weight (mg cos(θ)): \(F_f = \mu (mg \cos \theta)\)

Resolve the horizontal force

The horizontal force F can also be broken down into components parallel and perpendicular to the inclined plane: - Parallel component: \(F \cos \theta\) - Perpendicular component: \(F \sin \theta\)

Apply Newton's Second Law

Now we apply Newton's second law along the parallel direction, taking into account the horizontal force component, weight component, and the friction force: - \(ma_{\parallel} = F_{\parallel} - mg_{\parallel} - F_f\) Substitute the known values and the expressions derived in Steps 2, 3, and 4: \(m a_{\parallel} = (F \cos \theta) - (mg \sin \theta) - (\mu (mg \cos \theta))\)

Solve for acceleration

Now, solve for the acceleration along the plane (a_𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙). \(a_{\parallel} = \frac{(F \cos \theta) - (mg \sin \theta) - (\mu (mg \cos \theta))}{m}\) Plug in the given values of m, θ, F, and μ: \(a_{\parallel} = \frac{(302.3 \mathrm{~N} \cos 21.3^{\circ}) - (36.1 \mathrm{~kg} \cdot 9.81 \mathrm{~m/s^2} \sin 21.3^{\circ}) - (0.159 (36.1 \mathrm{~kg} \cdot 9.81 \mathrm{~m/s^2} \cos 21.3^{\circ}))}{36.1 \mathrm{~kg}}\) After calculating the expression, we find: \(a_{\parallel} \approx 2.95 \mathrm{~m/s^2}\) The acceleration of the wedge along the inclined plane is approximately \(2.95 \mathrm{~m/s^2}\).

Key concepts

Inclined Plane

An inclined plane is a flat surface that is tilted at an angle to the horizontal. It is one of the simplest forms of a simple machine and can be used to lift objects or make moving an object easier. When analyzing situations on inclined planes, we often consider how forces such as gravity and applied forces interact.
In this exercise, the plane is inclined at an angle of \(\theta = 21.3^{\circ}\). This means that instead of acting directly downward, the force of gravity is divided into two components: one parallel to the plane and one perpendicular. Understanding these components is crucial for solving problems related to inclined planes.

• The component parallel to the plane, \(mg \sin \theta\), pulls the object down the slope.
• The component perpendicular to the plane, \(mg \cos \theta\), affects the normal force exerted by the plane onto the object.

Understanding these two components helps to determine how other forces, such as friction and applied forces, will influence the object's movement on the inclined plane.

Kinetic Friction

Kinetic friction occurs when two surfaces move against each other. It acts to resist the relative motion between the surfaces and depends on the nature of the surfaces and the force pressing them together. The coefficient of kinetic friction \(\mu\) is a measure of how "rough" or "sticky" the two surfaces are.
In our exercise, the coefficient of kinetic friction between the wedge and the incline is 0.159. To calculate the frictional force \(F_f\) acting on the wedge, we use the normal force \(F_n\), which is the perpendicular component of the object's weight:

\[F_f = \mu F_n = \mu (mg \cos \theta)\]
This equation shows how the frictional force is directly proportional to the normal force and the coefficient of friction. Kinetic friction is an essential part of maintaining control over the movement of objects on surfaces and must be accounted for accurately, especially when calculating net forces and acceleration.

Acceleration Calculation

Newton's Second Law of Motion provides a way to calculate the acceleration of an object when forces are acting on it. The law states that the force on an object is equal to its mass times its acceleration \(F = ma\). For our exercise, we need to find the acceleration along the inclined plane.
To do this, we resolve all the forces acting along the direction of the incline:

• The parallel component of the applied force, \(F \cos \theta\).
• The gravitational force component pulling the object down the incline, \(mg \sin \theta\).
• The frictional force opposing the motion, \(F_f = \mu (mg \cos \theta)\).

By applying Newton's Second Law, these forces combine to give the equation:

\[ma_{\parallel} = F_{\parallel} - mg_{\parallel} - F_f\]
When we solve for \(a_{\parallel}\), we substitute the known values to find the acceleration of the wedge along the inclined plane:

\[a_{\parallel} = \frac{(302.3 \mathrm{~N} \cos 21.3^{\circ}) - (36.1 \mathrm{~kg} \cdot 9.81 \mathrm{~m/s^2} \sin 21.3^{\circ}) - (0.159 (36.1 \mathrm{~kg} \cdot 9.81 \mathrm{~m/s^2} \cos 21.3^{\circ}))}{36.1 \mathrm{~kg}}\]
The calculated acceleration \(a_{\parallel} \approx 2.95 \mathrm{~m/s^2}\) describes how quickly the wedge speeds up as it slides along the inclined plane.