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Chapter 8: Problem 39

A sled of mass \(70 \mathrm{kg}\) starts from rest and slides down a \(10^{\circ}\) incline \(80 \mathrm{m}\) long. It then travels for \(20 \mathrm{m}\) horizontally before starting back up an \(8^{\circ}\) incline. It travels \(80 \mathrm{m}\) along this incline before coming to rest. What is the magnitude of the net work done on the sled by friction?

Short Answer

Expert verified
The net work done on the sled by friction is given by: \[W_{total} = W_{friction(down)} + W_{friction(horizontal)} + W_{friction(up)}\] To find the total work done by friction, we first calculated the final velocity of the sled down the incline using the conservation of mechanical energy equation, then analyzed the motion on the horizontal surface and up the 8-degree incline using the work-energy principle. After finding the work done by friction during each section of the sled's motion, we added them up to obtain the total net work done by friction on the sled.

Step by step solution

01

Analyze the motion down the 10-degree incline

First, let's find the final velocity of the sled due to motion down the 10-degree incline using the conservation of mechanical energy. The initial gravitational potential energy (\(U_i\)) can be expressed as: \[U_i = mgh\] where \(m\) is the mass, \(g\) is acceleration due to gravity, and \(h\) is the height of the incline. The height of the incline can be calculated as: \[h = L\sin(\theta)\] where \(L\) is the length of the incline and \(\theta\) is the angle. In our case, \(L = 80 \mathrm{m}\) and \(\theta = 10^{\circ}\).
02

Calculate the final velocity of the sled down the incline

To find the final velocity of the sled (\(v_f\)), we are going to apply the conservation of mechanical energy equation: \[U_i = K_f - W_{friction}\] Since the sled starts from rest, its initial kinetic energy (\(K_i\)) is zero. The equation becomes: \[mgh = \frac{1}{2}mv_f^2 - W_{friction}\]
03

Analyze motion on the horizontal surface

As the sled moves 20 m horizontally, friction does work on the sled, decreasing its kinetic energy. We will consider this decrease in kinetic energy as \(K_{decrease}\), and it can be found using the work-energy principle: \[W_{friction} = -K_{decrease}\]
04

Analyze motion on the 8-degree incline

Now, the sled moves up an 8-degree incline. During this motion, the sled loses kinetic energy. The work-energy principle is still applicable here: \[W_{friction} = K_{decrease} - \Delta U_f\] where \(\Delta U_f\) is the change in gravitational potential energy as the sled moves up the incline.
05

Calculate the sum of work done by friction

Now, we will add the work done by friction during each section of the sled's motion: \[W_{friction(total)} = W_{friction(down)} + W_{friction(horizontal)} + W_{friction(up)}\] Since we have expressions for these quantities from steps 2, 3, and 4, we can plug them in to find the total work done by friction. First, by substituting \(mgh\) as given in step 1, we have the equation to find the final velocity of the sled (\(v_f\)): \[70g(80\sin(10^\circ)) = \frac{1}{2}(70)v_f^2 - W_{friction(down)}\] By solving for work done by friction for the horizontal and up components and plugging them in into the equation above, we finish the problem: \[W_{total} = W_{friction(down)} + W_{friction(horizontal)} + W_{friction(up)}\] Here, \(W_{total}\) represents the net work done on the sled by friction, which is the required answer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinetic Energy
Kinetic energy is a form of energy that an object possesses due to its motion. The faster an object moves, the greater its kinetic energy. For a sled moving down an incline, its initial kinetic energy is zero since it starts from rest. As it slides down, it gains speed.
The kinetic energy of an object can be calculated using the formula:
  • \[ K = \frac{1}{2}mv^2 \]
Here,
  • \( m \) is the mass of the object, and
  • \( v \) is its velocity.
When the sled moves horizontally and up the second incline, this energy is affected by friction, which reduces its speed and kinetic energy. Understanding kinetic energy is crucial when analyzing how energy is transformed as the sled changes between motion on different surfaces.
Potential Energy
Potential energy is the energy stored within an object due to its position relative to other objects. For the sled exercise, gravitational potential energy plays a vital role. The higher the sled's starting point on the incline, the more potential energy it has.
The potential energy due to gravity is given by:
  • \[ U = mgh \]
Where
  • \( m \) is the mass,
  • \( g \) is the acceleration due to gravity (approximated as \( 9.8\ \text{m/s}^2\)),
  • \( h \) is the height.
As the sled descends, this potential energy is converted into kinetic energy, helping the sled to move faster. Conversely, when the sled moves up the second incline, the kinetic energy is converted back into potential energy.
Work-Energy Principle
The work-energy principle is a fundamental concept in physics that relates the work done on an object to its energy changes. In this sled problem, the work-energy principle helps us understand how the sled's energy is transformed as it moves.
  • Work done by forces (like friction) results in a change in the sled's kinetic energy.
  • The principle is expressed by \( W = \Delta K \).
This indicates that the work done by friction, in particular, equates to the decrease in kinetic energy as the sled travels horizontally and up the second incline.
To solve the problem, apply this principle to each section of the sled's journey and account for all work done by friction. This allows for the calculation of the net work done by friction over the entire journey.
Friction
Friction is the force that opposes the motion of objects sliding against each other. It plays a crucial role in the sled exercise as it affects all stages of the sled's motion. When the sled slides down the first incline, friction works against its motion, reducing its net acceleration.
Key points about friction include:
  • It depends on the surfaces in contact.
  • It converts kinetic energy into heat.
  • The work done by friction can be calculated as the product of the friction force and the distance over which it acts.
In this problem, friction does negative work, taking energy away from the sled, and is responsible for the total reduction in mechanical energy as the sled moves across different surfaces.
Incline Plane
An incline plane is simply a flat surface tilted at an angle to the horizontal. It helps us analyze the sled problem by simplifying the forces involved. The most significant force acting on the sled is gravity, pulling it down the plane. However, this force can be split into two components: one parallel to the plane, which causes the sled to accelerate, and one perpendicular that is balanced by the normal force.What鈥檚 crucial about an incline plane:
  • The steeper the plane, the greater the component of gravitational force acting down the plane.
  • Energy conversion calculations (from potential to kinetic) often use the height associated with the incline, \( h = L\sin(\theta) \).
In the problem, understanding the angles and lengths of the incline allows us to calculate changes in energy and apply them to solve for friction's work, making the incline a straightforward way to parse out the sled's motion characteristics.

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