Problem 89 - University Physics with Modern Physics

Question

On an essentially frictionless, horizontal ice rink, a skater moving at 3.0 \(\mathrm{m} / \mathrm{s}\) encounters a rough patch that reduces her speed to 1.65 \(\mathrm{m} / \mathrm{s}\) due to a friction force that is 25\(\%\) of her weight. Use the work-energy theorem to find the length of this rough patch.

Step-by-Step Solution

Answer

The length of the rough patch is approximately 3.28 meters.

Step 1: Identify the Known Values

We start by jotting down the facts given in the problem. Initial speed of the skater, \( v_i = 3.0 \, \text{m/s} \); final speed, \( v_f = 1.65 \, \text{m/s} \); friction force is 25% of the skater's weight. The goal is to find the length of the rough patch, denoted as \( d \).

Step 2: Apply the Work-Energy Theorem

The work-energy theorem states that the work done by the forces is equal to the change in kinetic energy. The work done by the friction force (\( F_f \)) is \( F_f \cdot d \), and the change in kinetic energy (\( \Delta KE \)) is \( \frac{1}{2}m {v_f}^2 - \frac{1}{2}m {v_i}^2 \).

Step 3: Expression for Frictional Work Done

The work done by the friction force can be expressed as: \(-F_f \cdot d = \Delta KE \), where \( F_f = 0.25 \times mg \), with \( m \) being the skater's mass and \( g = 9.8 \, \text{m/s}^2 \).

Step 4: Set Up the Equation

Substitute \( F_f = 0.25mg \) into the equation: \(-0.25mg \cdot d = \frac{1}{2}m(1.65^2) - \frac{1}{2}m(3.0^2) \). Notice that the mass \( m \) cancels out from the equation.

Step 5: Simplify and Solve for \( d \)

Calculate the change in kinetic energy: \( \Delta KE = \frac{1}{2}(1.65^2) - \frac{1}{2}(3.0^2) \). Evaluate this value, then solve the equation for \( d \). The equation becomes \(-0.25g \cdot d = \Delta KE \), leading to \( d = \frac{-\Delta KE}{0.25g} \).

Step 6: Calculate the Length of the Rough Patch

Compute \( d = \frac{-\Delta KE}{0.25 \times 9.8} \). First, determine \( \Delta KE = \frac{1}{2}(1.65^2 - 3.0^2) \) which equals approximately \(-8.0475 \). Then compute \( d = \frac{8.0475}{0.25 \times 9.8} \), which is approximately 3.28 meters.

Key Concepts

Friction Force

Friction is a resistive force that acts to oppose the direction of motion of an object. It is due to the roughness of the surfaces in contact and can significantly affect motion. In the exercise, the skater experiences a friction force that is 25% of her weight when she glides over a rough patch on the ice rink. This friction acts against her motion, causing her to slow down.

The formula to determine this friction force is given by:

• Friction force ( F_f ) = 0.25 imes mg

Where mg represents the skater's weight, with m being her mass and g being the acceleration due to gravity (approximated as 9.8 m/s²).

This indicates that friction is directly proportional to weight. So, the heavier the skater, the larger the friction force she will encounter. Understanding friction is crucial as it converts kinetic energy into heat or sound, slowing down the skater on her path.

Kinetic Energy

Kinetic energy refers to the energy that an object possesses due to its motion. It is dependent on two factors: the mass of the object and the square of its velocity.

Kinetic energy is calculated using the formula:

• Kinetic energy (KE) = \( \frac{1}{2}m v^2 \)

Where m is the mass of the object, and v is its velocity.

In the context of the skater, her initial and final kinetic energies are critical in determining how much energy has been lost due to friction:

• Initial kinetic energy: \(\frac{1}{2}mv_i^2\)
• Final kinetic energy: \(\frac{1}{2}mv_f^2\)

The change in kinetic energy (ΔKE) helps us evaluate the energy transferred from motion into overcoming friction, which ultimately determines how far the skater travels along the rough patch. This calculation is integral to applying the work-energy theorem, linking the work done by friction to the change in kinetic energy.

Horizontal Motion

Horizontal motion refers to the movement along a straight line parallel to the horizon. In this exercise, the skater's horizontal motion takes place on an ice rink, where factors like friction come into play significantly to alter her speed.

In horizontal motion, the work-energy theorem comes into effect, especially when unbalanced forces like friction act on an object. This theorem ties together the work done by the forces to the change in the object's kinetic energy.

• The theorem stipulates that: Work done = Change in kinetic energy ( ΔKE )

The skater's motion is impeded by the friction force that acts on her. This interaction exemplifies how even in the absence of vertical forces (since it's horizontal), friction can slow down the movement. The work done by friction extracts energy from the skater’s motion, reducing her speed until she comes to a smoother surface.

This concept highlights the interconnected nature of physics concepts—understanding how they relate is key to deciphering problems involving motion and forces on a horizontal plane.