Question

In Fig. 5-38, object \(B\) weighs \(94.0 \mathrm{lb}\) and object \(A\) weighs \(29.0 \mathrm{lb} .\) Between object \(B\) and the plane the coefficient of static friction is \(0.56\) and the coefficient of kinetic friction is 0.25. (a) Find the acceleration of the system if \(B\) is initially at rest. (b) Find the acceleration if \(B\) is moving up the plane. ( \(c\) ) What is the acceleration if \(B\) is moving down the plane? The plane is inclined by \(42.0^{\circ}\).

Short answer

The acceleration of the system when \(B\) is at rest, moving up, and moving down the plane can be computed by performing the steps as shown above.

Step-by-step solution

Calculate the weight components

Split the weight of object \(B\) into two components: one parallel to the plane and another perpendicular to the plane. The weight components are given by \(W_{B\parallel} = W_B \sin(\theta)\) and \(W_{B\perp} = W_B \cos(\theta)\) where \(W_B = 94.0\, lb\) is the weight of object \(B\) and \(\theta = 42.0^{\circ}\) is the angle of inclination.

Determine the net force

The net force acting on the system when object \(B\) is at rest is equal to the difference between the force due to \(B\) moving down the incline and the static frictional force. The static frictional force can be calculated by multiplying the normal force (which is equal to \(W_{B\perp}\)) by the coefficient of static friction. Thus, \(F_{net} = W_{B\parallel} - \mu_s W_{B\perp}\) where \(\mu_s = 0.56\) is the coefficient of static friction.

Compute the acceleration for question (a)

The acceleration of the system can be found using Newton's second law \(F = ma\). Here, \(m\) is the total mass of the system, which is the sum of the weights of objects \(A\) and \(B\) divided by gravity \(g\). But since we are asked in 'lb' units, it is not necessary to divide by gravity. So, \(a = F_{net} / (W_A + W_B)\) where \(W_A = 29.0\, lb\) is the weight of object \(A\).

Calculate the acceleration for question (b)

If \(B\) is moving up the plane, the kinetic friction will act in the same direction as \(W_{B\parallel}\). The net force is now \(F_{net} = W_{B\parallel} + \mu_k W_{B\perp} - W_A\) where \(\mu_k = 0.25\) is the coefficient of kinetic friction. The acceleration is found using the same procedure as in step 3.

Determine the acceleration for question (c)

If \(B\) is moving down the plane, the kinetic friction acts in the opposite direction to \(W_{B\parallel}\). Now, \(F_{net} = W_{B\parallel} - \mu_k W_{B\perp} - W_A\). Again, calculate the acceleration as in step 3.

Key concepts

Static Friction

Static friction is a force that keeps an object at rest when it is placed on a surface. It acts to prevent the object from sliding down by opposing the component of gravitational force parallel to the surface. To better understand static friction:

• The maximum static friction is proportional to the normal force (the component of weight perpendicular to the surface).
• The static frictional force can be calculated using the formula: \( f_s = \mu_s \cdot N \), where \( \mu_s \) is the coefficient of static friction and \( N \) is the normal force.

In the given exercise, static friction plays a critical role when object \( B \) is initially at rest. The force resisting motion is calculated using the weight component perpendicular to the inclined plane to find the net force acting on the system.

Kinetic Friction

Kinetic friction comes into play once an object starts moving. Unlike static friction, kinetic friction does not vary; it has a constant value as long as the relative motion continues. Here are some important points:

• Kinetic friction is generally less than static friction for the same pair of surfaces.
• It can be calculated using the formula: \( f_k = \mu_k \cdot N \), where \( \mu_k \) is the coefficient of kinetic friction.

In this problem, when object \( B \) is moving up or down the plane, kinetic friction opposes the motion. If \( B \) is moving upwards, the kinetic friction adds to the force component trying to pull it down. When moving downwards, it acts against gravity, reducing the net force pulling \( B \) down the incline.

Newton's Second Law

Newton's Second Law of Motion tells us that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This law is mathematically expressed as \( F = ma \), where \( F \) is the net force, \( m \) is the mass, and \( a \) is the acceleration.
In the context of inclined plane problems:

• We find net forces that consider both gravitational components and frictional forces.
• The total weight of both objects \( A \) and \( B \) is used to determine their acceleration under different conditions.

For this exercise, once the net force is calculated whether from static or kinetic friction scenarios, it is divided by the total weight of objects \( A \) and \( B \) to find their acceleration, effectively applying Newton's Second Law.

Net Force

The net force is the overall force acting on an object or system, taking into account all contributing factors, such as gravity and friction. This single resulting force determines the object's motion or acceleration. Consider the following:

• Net force is the difference between the forward and opposing forces acting along the direction of interest.
• On an incline, it's calculated by subtracting opposing frictional forces from the downhill gravitational component.

For this exercise, understanding the net force involves calculating the gravitational forces on \( B \) and subtracting static or kinetic friction for different movement scenarios. Each of these scenarios requires careful evaluation of forces to determine the system's acceleration.

Inclined Plane

Inclined planes are flat surfaces tilted at an angle, and they present unique challenges due to the effects of gravity along both parallel and perpendicular components. These are some crucial aspects:

• The parallel component of an object's weight contributes to its potential motion down the plane.
• The perpendicular component affects the normal force, which is vital for calculating friction.

In inclined plane problems, like in this exercise, separate the weight of object \( B \) into parts that act along and against the plane. This separation is essential for understanding how different forces impact the acceleration and motion of objects, particularly when evaluating static and kinetic friction scenarios.