Question

Cookie Power To make a batch of cookies, you mix half a bag of chocolate chips into a bowl of cookie dough, exerting a \(21-\mathrm{N}\) force on the stirring spoon. Assume that your force is always in the direction of motion of the spoon. (a) What power is needed to move the spoon at a speed of \(0.23 \mathrm{~m} / \mathrm{s}\) ? (b) How much work do you do if you stir the mixture for \(1.5\) min?

Short answer

(a) 4.83 W is needed; (b) 434.7 J of work is done.

Step-by-step solution

Understand the Relationship

In physics, power (P) is the rate at which work is done. The formula for power in terms of force and velocity is given by:\[ P = F \cdot v \]where \( P \) is power, \( F \) is force, and \( v \) is velocity.

Calculate the Power

We are given that the force \( F = 21 \text{ N} \) and the velocity \( v = 0.23 \text{ m/s} \). Substitute these values into the formula for power:\[ P = 21 \text{ N} \times 0.23 \text{ m/s} = 4.83 \text{ W} \]

Understand the Relationship for Work

Work (W) is the energy transferred by a force acting through a distance. It is given by the product of power and time:\[ W = P \times t \]where \( W \) is work, \( P \) is power, and \( t \) is time.

Convert Time to Seconds

The time given is \( 1.5 \text{ minutes} \). Convert this time into seconds since the standard unit for time in physics is seconds.\[ 1.5 \text{ minutes} = 1.5 \times 60 \text{ seconds} = 90 \text{ seconds} \]

Calculate the Work Done

Use the power calculated in the second step and the time converted in the fourth step to find the work done:\[ W = 4.83 \text{ W} \times 90 \text{ s} = 434.7 \text{ J} \]

Key concepts

Power calculation

In physics, the concept of power helps us understand how quickly work is being done. Imagine trying to mix cookie dough with chocolate chips using a spoon. Here, power is the amount of energy you exert over time to keep the spoon moving. To calculate power, we use the formula:

• \( P = F \cdot v \)

This means power is the product of the force applied (\( F \)) and the velocity (\( v \)) at which the object moves. In our cookie mixing scenario, a force of 21 N is applied, moving the spoon at a speed of 0.23 m/s. By multiplying these, we determine that the necessary power to mix the dough is 4.83 Watts (W).
It's important to remember that power tells us about the efficiency of energy use. Higher power means quicker action, useful in many real-life situations.

Work-energy principles

The concept of work in physics is about energy transfer. When mixing cookie dough, work reflects the energy transferred by moving the spoon against the dough's resistance over a distance. The formula for work is:

• \( W = P \times t \)

Here, \( W \) is the work done, \( P \) is power, and \( t \) is the time for which the force is applied. So, if you were stirring for 1.5 minutes, converting that time to seconds (90 seconds) helps us use the standard units in our calculations.
The work done in our example is then found by multiplying the power (4.83 W) with the time (90 s), resulting in 434.7 Joules (J) of work.
Understanding work-energy principles helps us see how forces change energy between objects, important in fields ranging from engineering to everyday cooking!

Force and motion

Force and motion are fundamental in understanding the physics of movement, like stirring a spoon through dough. When you apply a force, such as 21 N on a spoon, it creates motion if the force is greater than the object's resistance.

Force

The force, in this case, acts in the direction of motion. So, if you push or pull in the same direction, energy efficiently transfers to keep it moving. Essentially, the stronger the force, the more power you exert.

Motion

Motion refers to the change in position of an object with time. Here, motion shows how your stirring changes the position of the dough and chocolate chips. Velocity, such as 0.23 m/s, indicates speed and direction.
Understanding how force impacts motion is crucial because it helps us predict and control how objects will move, ensuring efficient and desired outcomes during various activities like sports, machinery use, and even baking cookies!