Problem 60 - University Physics with Modern Physics

Question

You are applying a constant horizontal force \(\overrightarrow{F} = (-8.00\mathrm{N})\hat{\imath} + (3.00\mathrm{N})\hat{\jmath}\) to a crate that is sliding on a factory floor. At the instant that the velocity of the crate is \(\overrightarrow{\upsilon} = (3.20\mathrm{m/s})\hat{\imath} + (2.20 \mathrm{m/s})\hat{\jmath}\), what is the instantaneous power supplied by this force?

Step-by-Step Solution

Answer

The instantaneous power is \(-19.00\, \mathrm{W}\).

Step 1: Understand the Concept of Power

Power is the rate at which work is done, and in this context, it can be calculated using the dot product of the force vector and the velocity vector. The formula for instantaneous power is given by \( P = \overrightarrow{F} \cdot \overrightarrow{v} \), where \( \overrightarrow{F} \) is the force and \( \overrightarrow{v} \) is the velocity.

Step 2: Apply the Dot Product

Compute the dot product of the force vector \((-8.00\mathrm{N})\hat{\imath} + (3.00\mathrm{N})\hat{\jmath}\) and the velocity vector \((3.20\mathrm{m/s})\hat{\imath} + (2.20 \mathrm{m/s})\hat{\jmath}\). The dot product is calculated as follows: \( (-8.00 \times 3.20) + (3.00 \times 2.20) \).

Step 3: Calculate Each Component

Find the result of each multiplication: \((-8.00) \times (3.20) = -25.60\) and \((3.00) \times (2.20) = 6.60\).

Step 4: Sum the Components

Add the results of the components: \(-25.60 + 6.60 = -19.00\). This sum represents the instantaneous power.

Step 5: Interpret the Result

The result \(-19.00 \mathrm{W}\) is the instantaneous power supplied by the force. The negative sign indicates that the force is acting opposite to the direction of the velocity, possibly doing work on other systems or overcoming some form of resistance.

Key Concepts

Understanding the Dot Product

The dot product is a way to multiply two vectors, resulting in a scalar. It quantifies how much one vector goes in the direction of another. To calculate it, you multiply corresponding components and then sum them up. Consider vectors \( \overrightarrow{F} = (-8.00 \mathrm{N})\hat{\imath} + (3.00 \mathrm{N})\hat{\jmath} \) and \( \overrightarrow{v} = (3.20 \mathrm{m/s})\hat{\imath} + (2.20 \mathrm{m/s})\hat{\jmath} \). The dot product is calculated like this:

• Multiply the \( \hat{\imath} \)-components: \((-8.00) \times (3.20) = -25.60\)
• Multiply the \( \hat{\jmath} \)-components: \((3.00) \times (2.20) = 6.60\)

After finding the products, add them: \(-25.60 + 6.60 = -19.00\). Thus, the dot product here is \(-19.00\).
This result, a single number, tells us about the directional alignment and magnitude of interaction between the vectors.

The Role of the Force Vector

A force vector represents both the magnitude and direction of a force. In our example, \( \overrightarrow{F} = (-8.00 \mathrm{N})\hat{\imath} + (3.00 \mathrm{N})\hat{\jmath} \). Each component defines how much force is applied along a specific axis:

• -8.00 N in the direction of \( \hat{\imath} \), meaning force is applied left.
• 3.00 N in the direction of \( \hat{\jmath} \), indicating upward force.

These components impact how the crate moves. A vector allows us to consider forces separately along each axis, simplifying complex motion into understandable parts. Knowing the vector helps analyze forces in physics problems.

Understanding the Velocity Vector

The velocity vector indicates both the speed and direction of an object's motion. Here, it is given by \( \overrightarrow{v} = (3.20 \mathrm{m/s})\hat{\imath} + (2.20 \mathrm{m/s})\hat{\jmath} \). Each part describes movement:

• 3.20 m/s in \( \hat{\imath} \), indicating rightward movement.
• 2.20 m/s in \( \hat{\jmath} \), indicating upward movement.

The vector reflects how the crate travels over time. By using vectors, changes in speed or direction can be easily composed or decomposed into simpler parts. As with force, this approach reveals the complete picture of motion in a clear mathematical form.

Rate of Work and Instantaneous Power

Instantaneous power is the power at a specific moment, embodying the concept of the rate at which work is done. When a force acts on an object in motion, like the crate, power can be derived from the dot product of the force and velocity vectors: \[ P = \overrightarrow{F} \cdot \overrightarrow{v} = -19.00 \text{ W} \]This tells us how "quickly" force is doing work over time. Here, the negative result \(-19.00\) W indicates that the force's direction predominantly counters the crate's motion, such as overcoming friction or other forces. Thus, instantaneous power not only measures energy transformation but also gives insight into motion dynamics and resistance.