Problem 82 - University Physics with Modern Physics

Question

Pushing a Cat. Your cat "Ms." (mass 7.00 kg) is trying to make it to the top of a frictionless ramp 2.00 \(\mathrm{m}\) long and inclined upward at \(30.0^{\circ}\) above the horizontal. Since the poor cat can't get any traction on the ramp, you push her up the entire length of the ramp by exerting a constant \(100-\mathrm{N}\) force parallel to the ramp. If Ms. takes a running start so that she is moving at 2.40 \(\mathrm{m} / \mathrm{s}\) at the bottom of the ramp, what is her speed when she reaches the top of the incline? Use the work-energy theorem.

Step-by-Step Solution

Answer

Ms.'s speed at the top is approximately 6.58 m/s.

Step 1: Identify Known Values

Let's identify the known values from the problem:- Mass of the cat, \( m = 7.00 \, \text{kg} \).- Length of the ramp, \( d = 2.00 \, \text{m} \).- Angle of the incline, \( \theta = 30.0^{\circ} \).- Initial speed at the bottom, \( v_i = 2.40 \, \text{m/s} \).- Applied force parallel to the ramp, \( F = 100 \, \text{N} \).These values will be used throughout the calculation.

Step 2: Calculate Gravitational Force Along the Ramp

First, calculate the component of gravitational force acting along the ramp:\[ F_{\text{gravity}} = m \cdot g \cdot \sin \theta \]where \( g = 9.81 \, \text{m/s}^2 \) is the acceleration due to gravity.Substituting the values:\[ F_{\text{gravity}} = 7.00 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times \sin(30.0^{\circ}) = 34.335 \, \text{N} \]

Step 3: Determine Net Force and Work Done

Next, calculate the net force acting on the cat along the ramp:\[ F_{\text{net}} = F - F_{\text{gravity}} = 100 \, \text{N} - 34.335 \, \text{N} \]\[ F_{\text{net}} = 65.665 \, \text{N} \]The work done by this net force as the cat ascends the ramp is:\[ W_{\text{net}} = F_{\text{net}} \cdot d = 65.665 \, \text{N} \times 2.00 \, \text{m} = 131.33 \, \text{J} \]

Step 4: Use the Work-Energy Theorem

According to the work-energy theorem, the work done on the cat results in a change in kinetic energy:\[ \Delta KE = W_{\text{net}} \]The change in kinetic energy can also be denoted as:\[ \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2 = W_{\text{net}} \]Rearranging gives:\[ v_f^2 = \frac{2 W_{\text{net}}}{m} + v_i^2 \]

Step 5: Solve for Final Speed

Substitute the known values into the rearranged kinetic energy equation:\[ v_f^2 = \frac{2 \times 131.33 \, \text{J}}{7.00 \, \text{kg}} + (2.40 \, \text{m/s})^2 \]\[ v_f^2 = 37.523 + 5.76 \]\[ v_f^2 = 43.283 \]Finally, calculate \( v_f \):\[ v_f = \sqrt{43.283} \approx 6.58 \, \text{m/s} \]

Step 6: Conclusion

Therefore, the speed of the cat when it reaches the top of the ramp is approximately 6.58 m/s.

Key Concepts

Kinetic Energy

Kinetic energy is a fundamental concept in physics that refers to the energy a body possesses due to its motion. It is mathematically expressed as \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass of the object and \( v \) is its velocity. In the problem of Ms. the cat on the ramp, kinetic energy plays a crucial role as it considers the conversion of work done into energy that leads to changes in her speed. Initially, Ms. has some kinetic energy due to her running start with a speed of 2.40 m/s. As Ms. moves up the incline, work done by the net force transforms into kinetic energy, altering her speed at the top. Understanding how kinetic energy is affected by forces helps us deeply grasp how energy transformation occurs during motion.

Gravitational Force

Gravitational force is the force of attraction that every object with mass exerts on every other object. In our ramp problem, gravity pulls Ms. down the inclined plane. To calculate the component of gravitational force acting along the ramp, we use the formula: \[ F_{\text{gravity}} = m \cdot g \cdot \sin \theta \]Here, \( g \) represents the acceleration due to gravity, approximately \( 9.81 \text{ m/s}^2 \), and \( \theta \) is the angle of the incline. For Ms., we calculated the gravitational force to be approximately 34.335 N along the ramp. This force needs to be overcome to push Ms. up. It illustrates how important angles and alignment are in understanding forces on inclined surfaces.

Net Force

Net force is the sum of all forces acting on an object. It determines the extent of acceleration or deceleration. In the exercise with Ms. the cat, we determine net force along the ramp by subtracting gravitational force from the applied 100 N force: \[ F_{\text{net}} = F - F_{\text{gravity}} \]which equates to 65.665 N. This net force acts along the direction of motion. The reason net force is so important is it directly affects the work done on the cat, thereby altering her kinetic energy. Ultimately, net force provides insight into how different forces influence the movement and energy of objects, like pushing Ms. to the top of the ramp with changing speed.

Inclined Plane

An inclined plane is a flat surface tilted at an angle to the horizontal. It simplifies the analysis of forces acting on objects. In this physics problem, the inclined plane allows us to consider forces over a distance of 2 meters with an angle of 30 degrees. The orientation of Ms. on the incline determines the component of gravitational force acting parallel to the surface, making it essential to the calculation of both gravitational force and net force. By analyzing movement on an inclined plane, we understand the effect angles have on energy transformation, thereby predicting outcomes like changes in speed at the top of the incline.

Physics Problem Solving

Solving this physics problem involves applying the work-energy theorem, which relates work done on an object to its change in kinetic energy. The general method includes:

• Identifying and listing known values.
• Computing forces, like gravitational and net force.
• Calculating work done by these forces.
• Using these calculations to find changes in kinetic energy and consequently, changes in final speed.

Physics problem solving emphasizes the step-by-step logical deduction using core principles like energy conversion and force dynamics. As exemplified in determining Ms.'s speed at the top, it requires a systematic approach integrating mathematical and physical laws to achieve a solution.