Question

A spring with a spring constant of \(500 . \mathrm{N} / \mathrm{m}\) is used to propel a 0.500-kg mass up an inclined plane. The spring is compressed \(30.0 \mathrm{~cm}\) from its equilibrium position and launches the mass from rest across a horizontal surface and onto the plane. The plane has a length of \(4.00 \mathrm{~m}\) and is inclined at \(30.0^{\circ} .\) Both the plane and the horizontal surface have a coefficient of kinetic friction with the mass of \(0.350 .\) When the spring is compressed, the mass is \(1.50 \mathrm{~m}\) from the bottom of the plane. a) What is the speed of the mass as it reaches the bottom of the plane? b) What is the speed of the mass as it reaches the top of the plane? c) What is the total work done by friction from the beginning to the end of the mass's motion?

Short answer

Answer: To find the speeds of the mass at the bottom and top of the plane, use the mechanical energy conservation approach and calculate the initial potential energy stored in the spring, work done by friction on the horizontal surface and inclined plane, and the gravitational potential energy at the top of the plane. Then, calculate the kinetic energy at the bottom and top of the plane and find the corresponding speeds using the kinetic energy formula. The total work done by friction is the sum of the work done by friction on the horizontal surface and that on the inclined plane.

Step-by-step solution

1. Calculate the initial potential energy stored in the spring

First, we need to calculate the energy stored in the spring when it is compressed by 30 cm. The formula for potential energy of a spring is given by \(U_s=\frac{1}{2}kx^2\) where \(U_s\) is the potential energy, \(k\) is the spring constant, and \(x\) is the compression distance from the equilibrium position. Given \(k=500 \, N/m\) and \(x=0.30\, m\), we have: \(U_s=\frac{1}{2}(500 \, N/m)(0.3 \, m)^2\)

2. Calculate the work done by friction on the horizontal surface

We need to find the work done by friction on the horizontal surface covering a distance of 1.5 m. The work done by friction is given by \(W_{fric}=-f_k d\) where \(W_{fric}\) is the work done by friction, \(f_k=\mu_k m g\) is the kinetic friction force, \(\mu_k\) is the coefficient of kinetic friction, \(m\) is the mass of the object, \(g\) is the gravitational acceleration, and \(d\) is the distance covered. Given \(\mu_k=0.350,\) \(m=0.5\, kg,\) \(g=9.81 \, m/s^2,\) and \(d=1.5\, m\), we have: \(W_{fric}=-0.350(0.500\, kg)(9.81\, m/s^2)(1.50 \, m)\)

3. Determine the mechanical energy of the mass at the bottom of the plane

By the conservation of mechanical energy (neglecting air resistance), the sum of the initial potential energy stored in the spring and the work done by friction equals the final kinetic energy of the mass at the bottom of the plane. The kinetic energy is given by \(K_f=U_s+W_{fric}\) where \(K_f\) is the final kinetic energy.

4. Calculate the speed of the mass at the bottom of the plane

The speed of the mass can be found using the formula for kinetic energy: \(K_f=\frac{1}{2}mv^2\) where \(v\) is the speed of the mass. Solve for \(v\): \(v=\sqrt{\frac{2K_f}{m}}\) #b) Finding the speed of the mass at the top of the plane#

5. Calculate the gravitational potential energy at the top of the plane

We need to find the gravitational potential energy at the top of the inclined plane. The formula for gravitational potential energy is given by \(U_g=mgh\) where \(U_g\) is the potential energy due to gravity, \(h\) is the vertical height of the plane, and \(m\) and \(g\) are as defined in step 2. Given the length of the incline \(L=4.00\, m\) and the angle of inclination \(\theta=30.0^\circ\), we can calculate \(h\) as: \(h=L\sin\theta\) Using this, we can calculate the gravitational potential energy at the top of the inclined plane.

6. Calculate the work done by friction on the inclined plane

We need to find the work done by friction on the inclined plane covering a distance of 4.00 m. Using the formula mentioned in step 2, we have: \(W_{fric} = -\mu_k m g d \cos\theta\)

7. Determine the mechanical energy of the mass at the top of the plane

Again using the conservation of mechanical energy, we can write: \(K_f=K_i - U_g + W_{fric}\) where \(K_i\) is the initial kinetic energy of the mass at the bottom and \(K_f\) is the final kinetic energy at the top of the plane.

8. Calculate the speed of the mass at the top of the plane

Similar to step 4, find the speed of the mass at the top of the inclined plane using: \(v=\sqrt{\frac{2K_f}{m}}\) #c) Finding the total work done by friction from the beginning to the end of the mass's motion#

9. Calculate the total work done by friction

Simply add the work done by friction on the horizontal surface and that on the inclined plane: \(W_{total} = W_{fric_{horz}} + W_{fric_{incline}}\) The mechanical energy approach has helped us solve each part of the exercise step by step, allowing us to find the speed of the mass at the bottom and top of the inclined plane, and the total work done by friction throughout the motion.

Key concepts

Work-Energy Principle

Understanding the work-energy principle is fundamental when examining the motion of objects and the forces acting upon them. Essentially, this principle states that work done on an object results in a change in its kinetic energy. This is articulated in the equation, \( W_\text{net} = \Delta KE \), where \( W_\text{net} \) is the net work done on the object and \( \Delta KE \) is the change in kinetic energy.

In our scenario, the work done by the spring's released potential energy and the work done against friction both contribute to the net work that alters the mass's kinetic energy as it moves from rest up the inclined plane. It's crucial to differentiate between work done by conservative forces, like that from the spring, which is path-independent and stores or releases energy, and work done by non-conservative forces such as friction, which dissipates energy typically as heat.

Kinetic Friction

Kinetic friction plays a pivotal role in everyday physics, and it manifests when two surfaces slide against each other. The force of kinetic friction is calculated using the formula \( f_k = \mu_k N \), with \( \mu_k \) representing the coefficient of kinetic friction and \( N \) the normal force—essentially the force perpendicular to the surfaces in contact.

• The work done by this frictional force, as seen in our example, is opposite to the direction of motion, resulting in a loss of mechanical energy.
• The equation \( W_{fric}=-f_k d \) showcases that work is dependent on the distance \( d \) over which the force acts.
• We must remember that kinetic friction depends on the types of surfaces in contact and is independent of the speed of the sliding objects.

This is critical in problems involving motion across different surfaces, where kinetic friction can drastically impact the object's speed and acceleration.

Conservation of Mechanical Energy

The conservation of mechanical energy is a powerful concept in physics that explains how energy is transferred, but not lost, within an isolated system. In an ideal scenario without any non-conservative forces (such as friction or air resistance), the sum of an object's kinetic energy (KE) and potential energy (PE) remains constant.

The equation \( KE_i + PE_i = KE_f + PE_f \) holds true, where indices \( i \) and \( f \) denote initial and final states, respectively. In real-world applications, however, one must account for energy losses, usually through work done by non-conservative forces.

In our example, the mechanical energy of the mass is not conserved because of the work done by friction. As such, we must adjust the basic conservation equation to include the work of friction: \( KE_i + PE_i + W_{fric} = KE_f + PE_f \).

Inclined Plane Physics

An inclined plane, a flat surface tilted at an angle, simplifies the problem of moving objects up a height, revealing insightful physics when applied in exercises. Several factors should be considered when analyzing an object moving on an inclined plane:

• The normal force changes as it's equal to the component of the object's weight perpendicular to the plane: \( N = m g \cos(\theta) \), where \( \theta \) is the angle of the plane's incline.
• Gravitational potential energy on an inclined plane can be modeled as \( U_g = m g h \), where \( h = L \sin(\theta) \), with \( L \) being the length of the slope.
• Frictional forces also adapt to the incline, as they are dependent on the normal force, which varies with the plane's angle.

In the context of our problem, the inclined plane adds complexity to calculating the work of friction and determining the mechanical energy of the mass at various points along its path.