Question

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, (b) 434.7 J

Step-by-step solution

Calculate Power

To find the power required, use the formula for power: $P=F\cdot v$, where $P$ is power, $F$ is the force applied, and $v$ is the velocity of the object. Here, $F=21$ N and $v=0.23$ m/s. Substitute these values into the equation: $P=21\times 0.23=4.83$ W.

Convert Time to Seconds

The problem states the time as 1.5 minutes. Convert this time into seconds because work is typically calculated in terms of seconds. There are 60 seconds in a minute, so $1.5$ minutes is equivalent to $1.5\times 60=90$ seconds.

Calculate Work Done

To find the work done, use the equation $W=P\times t$, where $W$ is work, $P$ is the power, and $t$ is the time in seconds. Substitute the values $P=4.83$ W and $t=90$ s into the equation: $W=4.83\times 90=434.7$ J.

Key concepts

Power calculation

Power in physics refers to the rate at which work is done or energy is transferred over time. When calculating power, it's essential to remember that it gives an idea of how quickly work is being performed. The formula used to compute power in terms of force and velocity is:

• $P=F\cdot v$

where $P$ represents power, $F$ is the force applied, and $v$ is the velocity of the object.

For the cookie dough mixing scenario, a force of 21 Newtons is applied to the spoon, moving at a velocity of 0.23 meters per second. By substituting these values into the power formula, the power needed is:

• $P=21\times 0.23=4.83\mathrm{W}$

This result indicates that 4.83 watts of power are required to continuously stir the dough at that rate. Power calculation is crucial because it helps us understand the efficiency and effectiveness of any mechanical system or physical activity.

Work done

The concept of work in physics is commonly understood as the energy transferred by a force moving an object over a distance. However, in this case, as we are dealing with a constant velocity of the spoon, work is calculated through the power and time formula:

• $W=P\times t$

Here, $W$ is the work done, $P$ is the power, and $t$ is the duration of time for which the force is applied, in seconds.

Before we can calculate the work done, we must convert the stirring time from minutes to seconds because the standard unit for time in such equations is seconds. Given 1.5 minutes of stirring time:

• The equivalent time in seconds is $1.5\times 60=90$ seconds.

The work done by stirring the mixture is:

• $W=4.83\times 90=434.7\mathrm{J}$

This calculation shows that 434.7 joules of energy are needed to perform the task of stirring with the given force and time.

Force and velocity

Understanding the relationship between force and velocity is crucial when analyzing physical systems. Force is defined as any interaction that, when unopposed, will change the motion of an object. Velocity, on the other hand, measures how fast an object is moving in a particular direction. When a force is applied at a constant velocity, like in the case of stirring cookie dough, it suggests that the net force is balanced by resistive forces, leading to steady motion.

In practical scenarios, the combination of force applied in the direction of motion and velocity can be used to determine the effectiveness of performance through power. If a larger force were applied at the same velocity, more work would be done in the same amount of time.

• If the velocity increased with the same force, this would also imply more power is required.

By maintaining a constant velocity, the task of stirring becomes stable, showcasing how the concepts of force and velocity interplay to ensure consistent mechanical work is accomplished smoothly and efficiently.