Problem 65 - University Physics with Modern Physics

Question

CP A 20.0 -kg crate sits at rest at the bottom of a 15.0 -m-long ramp that is inclined at \(34.0^{\circ}\) above the horizontal. A constant horizontal force of 290 \(\mathrm{N}\) is applied to the crate to push it up the ramp. While the crate is moving, the ramp exerts a constant frictional force on it that has magnitude 65.0 \(\mathrm{N}\) . (a) What is the total work done on the crate during its motion from the bottom to the top of the ramp? (b) How much time does it take the crate to travel to the top of the ramp?

Step-by-Step Solution

Answer

(a) The total work done is determined by calculating net force and multiplying by the ramp length. (b) Use kinematics to find the time given the acceleration.

Step 1: Analyze the forces on the crate

Identify the forces acting on the crate: the applied horizontal force (290 N), the gravitational force acting downward, the frictional force (65.0 N), and the normal force perpendicular to the ramp's surface. The gravitational force can be calculated as \( F_g = mg \), where \( m = 20.0 \, \text{kg} \) and \( g = 9.81 \, \text{m/s}^2 \).

Step 2: Determine the component of the applied force along the ramp

The applied force of 290 N is horizontal, but we need to find the component along the ramp: \( F_{app, \text{parallel}} = F_{app} \cdot \cos(\theta) \), where \( \theta = 34.0^{\circ} \). Compute this component to use in further calculations.

Step 3: Calculate the gravitational force component parallel to the ramp

The component of gravitational force acting along the ramp is \( F_{g, \text{parallel}} = mg \cdot \sin(\theta) \). Use \( m = 20.0 \, \text{kg} \) and \( \theta = 34.0^{\circ} \) to find \( F_{g, \text{parallel}} \).

Step 4: Find the net force along the ramp

The net force along the ramp, \( F_{net} \), is given by the sum of forces along the ramp: \( F_{net} = F_{app, \text{parallel}} - F_{g, \text{parallel}} - F_{friction} \), where \( F_{friction} = 65.0 \, \text{N} \). Calculate \( F_{net} \).

Step 5: Calculate the work done by the net force

The work done by the net force is given by \( W = F_{net} \cdot d \), where \( d = 15.0 \, \text{m} \) is the length of the ramp. Substitute the value of \( F_{net} \) to find the total work done on the crate.

Step 6: Calculate the acceleration of the crate

Using Newton's second law, \( F_{net} = ma \), solve for the acceleration \( a \) of the crate, where \( m = 20.0 \, \text{kg} \).

Step 7: Determine the time taken to reach the top of the ramp

Use the kinematic equation \( d = \frac{1}{2} a t^2 \) to solve for the time \( t \), given \( d = 15.0 \, \text{m} \) and the calculated acceleration \( a \).

Key Concepts

Force Analysis

Understanding the forces acting on an object is crucial in physics. In our scenario involving the crate, several forces are at play:

• Applied Force: A horizontal force of 290 N pushes the crate up the ramp.
• Gravitational Force: This force acts downward and is calculated as the product of mass (20.0 kg) and acceleration due to gravity (9.81 m/s²), resulting in a gravitational force downward.
• Frictional Force: Acts in the opposite direction to the crate's movement, with a magnitude of 65.0 N.
• Normal Force: Acts perpendicular to the ramp's surface, balancing the component of the gravitational force perpendicular to the ramp.

To solve related problems, consider both the direction and magnitude of these forces. Force analysis involves breaking down forces into components, especially in inclined motion, to understand their contribution to net force.

Frictional Force

Frictional forces are a core element when considering objects moving across surfaces. It is the resistance force that opposes the relative motion of an object. In the case of the crate moving up the inclined ramp, friction acts to slow it down. The frictional force here is a constant 65.0 N, opposing the motion. Frictional forces depend on:

• The nature of surfaces in contact.
• The normal force pressing the surfaces together. This is perpendicular to the direction of motion and impacts the frictional force significantly.

When calculating net force, always include the frictional force as it reduces the effectiveness of the applied force, thus impacting acceleration and travel time.

Kinematics

Kinematics is the study of motion without regarding the forces causing it. For the crate, we are interested in how it moves up the ramp:Using kinematic equations, we can determine how long it takes to reach the top:

• The primary equation here is: \[ d = \frac{1}{2} a t^2 \]where \(d\) is the distance (15.0 m), \(a\) is the acceleration obtained from force analysis, and \(t\) is the time to find.
• Solve this equation for \(t\) to find the duration of travel.

Understanding these principles helps predict motion accurately and interpret the effects of different forces acting on the crate.

Newton's Laws

Newton's Laws of Motion are fundamental to understanding force and motion.- First Law (Inertia): States that an object will remain at rest or in uniform motion unless acted on by an external force. Since the crate is initially at rest, a force is necessary to push it up the ramp.- Second Law (F=ma): Relates force, mass, and acceleration. We use this law to calculate the crate’s acceleration: \[ F_{net} = ma \]Net force (\(F_{net}\)) is the sum of all forces acting along the ramp, solving for acceleration \(a\).- Third Law (Action-Reaction): Every action has an equal and opposite reaction. The crate exerts a force on the ramp, and the ramp exerts an equal force in the opposite direction.Understanding these laws aids in developing a comprehensive view of motion and forces, enhancing problem-solving skills in physics.