Problem 18 - Physics for Scientists and Engineers with Modern Physics

Question

(II) The crate shown in Fig. 33 lies on a plane tilted at an angle \(\theta=25.0^{\circ}\) to the horizontal, with \(\mu_{\mathrm{k}}=0.19\) . (a) Determine the acceleration of the crate as it slides down the plane. (b) If the crate starts from rest 8.15 \(\mathrm{m}\) up the plane from its base, what will be the crate's speed when it reaches the bottom of the incline?

Step-by-Step Solution

Answer

The acceleration is 2.14 m/s², and the speed at the bottom is 5.91 m/s.

Step 1: Analyze the forces acting on the crate

First, identify the forces acting on the crate. The forces include gravitational force, normal force, and frictional force. The component of gravitational force parallel to the incline can be written as \( mg\sin\theta \), and the perpendicular component is \( mg\cos\theta \), where \( m \) is the mass of the crate. The normal force \( N \) is equal to \( mg\cos\theta \) because it counteracts the perpendicular component of gravity.

Step 2: Calculate the force of kinetic friction

The frictional force opposing the motion is the kinetic friction, calculated as \( f_k = \mu_k N \). Substitute the normal force: \( f_k = \mu_k mg\cos\theta \), with \( \mu_k = 0.19 \) and \( \theta = 25.0^{\circ} \). Thus, \( f_k = 0.19 \times mg\cos(25.0^{\circ}) \).

Step 3: Apply Newton's second law along the plane

Using Newton's second law \( F_{net} = ma \), the net force along the incline is \( mg\sin\theta - f_k = ma \). Substitute the expression for \( f_k \) into the equation: \( mg\sin\theta - 0.19 \times mg\cos(25.0^{\circ}) = ma \). Simplify to find the acceleration: \( a = g(\sin\theta - 0.19\cos(\theta)) \).

Step 4: Substitute numerical values to find acceleration

Calculate the acceleration \( a \). Using \( g = 9.81 \, \mathrm{m/s^2} \), \( \theta = 25.0^{\circ} \): \( a = 9.81(\sin(25.0^{\circ}) - 0.19\cos(25.0^{\circ})) \). Compute to get \( a = 2.14 \, \mathrm{m/s^2} \).

Step 5: Use kinematic equations to find final speed

Since the crate starts from rest, its initial velocity \( u = 0 \). Use the kinematic equation \( v^2 = u^2 + 2as \). Here, \( s = 8.15 \, \mathrm{m} \) is the distance along the plane, and \( a = 2.14 \, \mathrm{m/s^2} \). Substitute the values: \( v^2 = 0 + 2 \times 2.14 \times 8.15 \).

Step 6: Calculate the final speed

Perform the calculation from Step 5: \( v^2 = 34.892 \), hence \( v = \sqrt{34.892} \). Calculate \( v \), which gives the final speed as \( 5.91 \, \mathrm{m/s} \).

Key Concepts

Newton's Second Law

Newton's Second Law is a fundamental principle in physics that helps us understand how force and motion are related. At its core, this law states that the acceleration (\(a\)) of an object is directly proportional to the net force (\(F_{net}\)) acting upon it, and inversely proportional to its mass (\(m\)). This can be mathematically expressed as:

• \[F_{net} = ma\]

In simpler terms, if you push or pull something, the object will accelerate in the direction of the force you apply. However, how much it accelerates depends on the size of the force and the mass of the object.

In the context of this problem, the crate on an incline experiences several forces:

• Gravitational Force: Pulls the crate downwards. It has components parallel (\(mg\sin\theta\)) and perpendicular (\(mg\cos\theta\)) to the incline.
• Normal Force: Acts perpendicular to the incline and balances the perpendicular component of gravity.
• Kinetic Friction: Opposes the motion and acts along the surface of the incline.

Newton's Second Law is crucial here as it allows us to calculate the net force parallel to the incline, which in turn gives us the crate's acceleration down the slope.

Kinetic Friction

Kinetic friction is a type of frictional force that occurs when an object slides over a surface. It opposes the direction of motion and acts to slow down the moving object. The magnitude of kinetic friction (\(f_k\)) depends on two key factors:

• The coefficient of kinetic friction (\(\mu_k\)), which is a measure of how "sticky" the surfaces in contact are.
• The normal force (\(N\)), which is the perpendicular force exerted by a surface on the object resting on it.

The formula to calculate kinetic friction is given by:

• \[f_k = \mu_k N\]

For this particular problem, once we know the normal force (\(N = mg\cos\theta\)) exerted on the crate as it slides down the incline, we can easily calculate the force of kinetic friction as:

\(f_k = \mu_k mg\cos\theta\).

This force will act uphill, opposite to the gravitational force component pulling the crate downhill, hence reducing the overall acceleration of the crate.

Motion Along an Incline

Understanding motion along an incline involves analyzing how gravity and other forces impact an object's motion on a slope. When dealing with inclines, we often decompose forces into components that run parallel and perpendicular to the slope. This helps to predict and resolve the motion behavior effectively.

1. **Inclined Planes and Angle of Inclination**:
The inclination angle (\(\theta\)) significantly affects the motion on a slope. The steeper the angle, the more gravity will work to accelerate the object along the incline.

2. **Calculating Acceleration and Speed**:

• The primary forces to consider are the gravitational force, the normal force, and any frictional forces.
• The net force along the incline determines the acceleration (\(a = g(\sin\theta - \mu_k\cos\theta)\)).
• Using kinematic equations, the final speed of an object starting from rest can be derived from its acceleration and the distance down the incline.

For instance, if a crate starts sliding from rest at the top of an inclined plane, you can use the equation:

• \[v^2 = u^2 + 2as\]

where:

• \(u\) is the initial velocity,
• \(s\) is the distance down the slope,
• and \(v\) is the final speed.

Applying these principles helps us not just understand, but predict the motion of objects on inclined surfaces!