Question

An 8.00-kg block of ice, released from rest at the top of a 1.50-m-long frictionless ramp, slides downhill, reaching a speed of 2.50 m/s at the bottom. (a) What is the angle between the ramp and the horizontal? (b) What would be the speed of the ice at the bottom if the motion were opposed by a constant friction force of 10.0 N parallel to the surface of the ramp?

Short answer

(a) The angle is about 23.6°. (b) The speed with friction is about 1.87 m/s.

Step-by-step solution

Calculate Gravitational Potential Energy at the Top

The potential energy at the top of the ramp can be calculated using the formula \( PE = mgh \), where \( m = 8.00 \, \text{kg} \), \( g = 9.81 \, \text{m/s}^2 \), and \( h \) is the vertical height of the ramp which will be determined later.

Determine Kinetic Energy at the Bottom

The kinetic energy at the bottom of the ramp is given by \( KE = \frac{1}{2}mv^2 \), where \( m = 8.00 \, \text{kg} \) and \( v = 2.50 \, \text{m/s} \). Calculate \( KE \) to find the energy converted from potential to kinetic.

Apply Energy Conservation

Without friction, energy conservation states \( PE_{\text{top}} = KE_{\text{bottom}} \). The height \( h \) can be found by equating \( mgh = \frac{1}{2}mv^2 \). This simplifies to \( h = \frac{v^2}{2g} \). Substitute the known values to find \( h \).

Find the Angle Using Trigonometry

The length of the ramp is 1.50 m, and the height \( h \) is known from the previous step. Use \( \sin(\theta) = \frac{h}{\text{length of ramp}} \) to solve for \( \theta \). Calculate \( \theta \) using inverse sine.

Set Up Work-Energy Principle with Friction

When friction is present, the work done by friction is \( W_f = f \cdot d \), where \( f = 10.0 \, \text{N} \) and \( d = 1.50 \, \text{m} \). The energy equation becomes \( PE_{\text{top}} = KE_{\text{bottom}} + W_f \).

Calculate the New Speed with Friction

Rearrange the work-energy equation to find the new velocity: \[ \frac{1}{2}mv'^2 = mgh - f \cdot d \]. Solve for \( v' \) by substituting known values to find the speed at the bottom with friction.

Key concepts

Conservation of Energy

The conservation of energy is a fundamental principle in physics. It tells us that energy cannot be created or destroyed. Instead, it can only be transformed from one form to another. In this problem, we start with gravitational potential energy at the top of the ramp. As the block of ice slides down, this potential energy turns into kinetic energy.
At the very top, the ice has high potential energy because of its height. When the ice reaches the bottom, this energy becomes kinetic energy, which is associated with motion.

• Potential Energy (PE): Energy stored due to an object's position, calculated using \( PE = mgh \), where \( h \) is the height.
• Kinetic Energy (KE): Energy of motion, calculated using \( KE = \frac{1}{2}mv^2 \), where \( v \) is velocity.

By equating the potential energy at the top with the kinetic energy at the bottom, we ensure that the total energy remains constant in the absence of friction.

Kinematics

Kinematics is the branch of mechanics that describes the motion of objects, without referring to the forces that cause the motion. In this problem, kinematics helps us understand the ice block's movement down the ramp, particularly its change in speed from rest to 2.50 m/s.
Kinematics tells us how to describe motion with parameters such as displacement, velocity, and acceleration. For instance, knowing the distance the block travels down the ramp (1.50 m) and its final speed (2.50 m/s) helps determine its acceleration and the forces involved, assuming other influencing factors are given.

• Displacement: The block moves a distance of 1.50 m down the ramp.
• Velocity: Ranges from 0 m/s at rest to 2.50 m/s at the bottom.
• Acceleration: Determined by changes in speed and direction over time.

Knowledge of kinematics is crucial for calculating the other dynamic properties needed to solve the problem.

Trigonometry

Trigonometry involves the study of triangles, relationships between them, and the angles therein. This problem provides an opportunity to apply trigonometric principles to real-world physics. To find the angle of the ramp, we use the sine function from trigonometry.
Given the height \( h \) of the ramp and its length (1.50 m), the sine of the angle \( \theta \) of the ramp is given by \( \sin(\theta) = \frac{h}{\text{length of ramp}} \). By rearranging and solving for \( \theta \), we use inverse sine (\( \arcsin \)) to find the specific angle in degrees or radians.

• Sine function (\( \sin \)): Ratio of opposite side over hypotenuse in a right triangle.
• Inverse sine (\( \arcsin \)): Used to calculate the angle when the sine ratio is known.

Trigonometry aids in understanding the physical setup and determining critical angles for problem-solving.

Work-Energy Principle

The Work-Energy Principle bridges the concepts of work and energy, showing how forces acting over distances alter the energy of a system. In our problem, this principle explains what happens when the block moves along the ramp with friction.
The work done by friction changes the energy balance. Initially, all the gravitational energy converts to kinetic energy, but with friction, some is converted into heat due to work done against movement.

• Work (\( W \)): Energy transfer when a force moves an object over a distance, expressed as \( W = f \cdot d \).
• Friction Work: Work done by friction is simply the friction force times the displacement along the ramp.

The work-energy principle thus modifies the energy equation to factor in energy loss due to friction, adjusting the kinetic energy at the bottom of the ramp accordingly.

Friction

Friction is the resistive force opposing motion between surfaces. It plays a crucial role in the extended part of this problem, where the ramp is no longer frictionless. The friction force is a constant 10.0 N here.
Friction does work against the motion, making the ice block lose some of the energy it would otherwise convert to kinetic energy. This loss affects the final speed of the ice block.

• Static vs. Kinetic Friction: Focus here is on kinetic friction, impacting motion.
• Energy Loss: Friction converts some mechanical energy into heat, reducing kinetic energy.

Understanding how friction alters the energy dynamics of moving objects is vital when calculating their final speeds after traveling along surfaces with resistance.