Problem 23 - University Physics with Modern Physics

Question

A sled with mass 8.00 \(\mathrm{kg}\) moves in a straight line on a frictionless horizontal surface. At one point in its path, its speed is \(4.00 \mathrm{m} / \mathrm{s} ;\) after it has traveled 2.50 \(\mathrm{m}\) beyond this point, its speed is 6.00 \(\mathrm{m} / \mathrm{s}\) . Use the work-energy theorem to find the force acting on the sled, assuming that this force is constant and that it acts in the direction of the sled's motion.

Step-by-Step Solution

Answer

The constant force acting on the sled is 32 N.

Step 1: Understand the Concepts

The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. Therefore, we can write that \(W = \Delta KE = KE_f - KE_i\), where \(KE_f\) and \(KE_i\) are the final and initial kinetic energies, respectively. Since work \(W\) done by a constant force is \(F \cdot d\), where \(F\) is the force and \(d\) is the displacement, we can relate this to the work-energy theorem.

Step 2: Calculate Initial and Final Kinetic Energy

The initial kinetic energy (\(KE_i\)) is calculated using the formula \(KE = \frac{1}{2}mv^2\). Substitute mass \(m = 8.00 \text{ kg}\) and initial speed \(v_i = 4.00 \text{ m/s} \) to get \(KE_i = \frac{1}{2} \times 8 \times 4^2 = 64 \text{ J}\). The final kinetic energy (\(KE_f\)) is calculated similarly with final speed \(v_f = 6.00 \text{ m/s}\), giving \(KE_f = \frac{1}{2} \times 8 \times 6^2 = 144 \text{ J}\).

Step 3: Determine the Change in Kinetic Energy

Now that we have the initial and final kinetic energies, calculate the change in kinetic energy using \(\Delta KE = KE_f - KE_i\). Substituting the values, \(\Delta KE = 144 \text{ J} - 64 \text{ J} = 80 \text{ J}\).

Step 4: Relate Work Done to Force

The work done is equal to the change in kinetic energy, so \(W = \Delta KE = 80 \text{ J}\). From the work-energy principle \(W = F \cdot d\), and since \(d = 2.50 \text{ m}\), we can find \(F\) by rearranging the equation to \(F = \frac{W}{d}\).

Step 5: Calculate the Force Acting on the Sled

Substitute the known values into the equation \(F = \frac{80}{2.50}\), which gives \(F = 32 \text{ N}\). This is the constant force acting on the sled in the direction of its motion.

Key Concepts

Kinetic Energy

Kinetic energy is a type of energy that an object possesses due to its motion. It’s a vital part of understanding dynamics and how forces influence the movement of objects. The formula for kinetic energy is given by:

• \( KE = \frac{1}{2}mv^2 \)

where:

• \( KE \) is the kinetic energy,
• \( m \) is the mass of the object,
• \( v \) is the velocity of the object.

The kinetic energy changes whenever there is a change in either the mass or the velocity of an object. However, in most systems, the mass remains constant, and any change in kinetic energy results primarily from a change in velocity.
In the provided exercise, the sled's kinetic energy changed as its speed increased from \( 4.00 \text{ m/s} \) to \( 6.00 \text{ m/s} \). We calculated this change by evaluating the initial and final kinetic energies using the speeds given, highlighting the effect of increased velocity on kinetic energy.

Constant Force

In physics, a constant force is an unchanging force that acts on an object over time. Unlike variable forces, constant forces maintain the same magnitude and direction. This makes calculations simpler and is particularly useful when discussing straightforward situations.
According to the work-energy theorem, the work done (\( W \)) by a constant force is equal to the product of the force (\( F \)) and the displacement (\( d \)) in the direction of the force. Mathematically, this can be expressed as:

• \( W = F \cdot d \)

When applying the concept of force in our original exercise, we linked the constant force to the change in kinetic energy over a given displacement. Since the sled’s speed increased as it moved, the force must have been applied in the direction of the sled’s movement. By calculating the work done using the change in kinetic energy, we determined the constant force acting on the sled to be \( 32 \text{ N} \). This information is crucial, as it helps determine how much force was applied to affect the sled's motion on a frictionless surface.

Displacement

Displacement refers to the overall change in an object’s position. It is not merely the distance traveled but the net change from the starting to the ending position in a particular direction. Displacement is a vector quantity, which means it has both magnitude and direction.
In the context of the work-energy relationship, displacement plays a crucial role. It is the parameter that combines with force to give us work done through the term \( F \cdot d \).
For the sled in the original exercise, the displacement was given as \( 2.50 \text{ m} \). This was the distance over which the force acted, causing the sled's speed to increase and its kinetic energy to change. By knowing this distance, we were able to calculate how much work was done on the sled, which enabled us to deduce the magnitude of the force responsible for the sled’s acceleration.