Question

If the rate at which work is done on an object is increased, does the power supplied to that object increase, decrease, or stay the same?

Short answer

The power supplied to the object increases.

Step-by-step solution

Understand the Relationship between Work and Power

Power is defined as the rate at which work is done. Mathematically, power \( P \) is expressed as \( P = \frac{W}{t} \), where \( W \) is the work done and \( t \) is the time taken to do that work.

Analyze the Effect of Increased Work Rate

If the rate at which work is done increases, it means that more work is being accomplished in a given amount of time. This translates to an increase in the numerator \( W \) of the power formula, assuming the time \( t \) remains constant or decreases.

Determine the Impact on Power

Since power is the quotient of work done and time, an increase in the work done over the same or shorter period results in an increase in power. Thus, the power supplied to the object increases.

Key concepts

Work-Rate Relationship

In physics, understanding the connection between work and rate is crucial when discussing power. Work refers to the energy transferred when an object is moved over a distance by a force. It's calculated as the product of force and distance: \( W = F \times d \), where \( W \) is the work, \( F \) is the force applied, and \( d \) is the distance moved in the direction of the force. The term 'rate' in this context refers to how quickly this work is being accomplished. When we talk about the work-rate relationship, we're interested in how fast this energy transfer is taking place over time. This is the essence of power. A clear correlation exists: the faster the rate at which work is done, the higher the power. For example, if two cars are pushed over the same distance, but one takes less time, it means the work rate is higher, indicating greater power.

Equation for Power

The equation for power provides a concise way to calculate how much work is done over a specific time frame. It's expressed as:\[ P = \frac{W}{t} \]where \( P \) represents power, \( W \) is the work done, and \( t \) is the time taken. This equation highlights a simple division: the total work is evenly distributed over the time period it takes to perform that work.To understand this equation better, consider a scenario where you need to lift a stack of books onto a shelf. The amount of energy (or work) required is the same whether you take a full minute or just a few seconds to complete the task. However, if you accomplish the task in less time, you are working at a higher rate, resulting in higher power output.This formulation, thus, allows us to quantify power in units of Watts, where one Watt is equal to one joule per second.

Impact of Time on Power

Time, as a factor in the power equation, plays a significant role in determining the power output. The relationship between time and power is inherently inverse. As the time taken to do a given amount of work decreases, the power required increases, assuming the work done stays constant.Consider, for instance, running versus walking up a flight of stairs:- Running up the stairs requires the work to be done in a shorter time, thereby increasing power.- Walking slower takes more time, hence reducing power.When time decreases, the denominator in the power equation \( P = \frac{W}{t} \) becomes smaller, leading to a larger value of power. Conversely, if time increases, the power diminishes, assuming no change in the work done. This fundamental insight underlines the role of time in real-world scenarios like athletic performance and machine efficiency.