# Calculating Work Done to Move Arm from Position 1 to 2

• Haroon Pasha
In summary, the work required to move the arm from position 1 to position 2 is -0.21mg(cos(30)-1), where m is the mass of the arm and g is the acceleration due to gravity. To find the force applied to move the arm, we use the equation W = F*delta x*cos(angle). By treating the arm as a single point mass attached to a rigid, massless rod, we can use the line integral to find the work done by gravity. Using trigonometry, we can also determine the exact height the arm has moved, making the calculation more accurate. Thank you Quasar987 and AM for your helpful explanations.

#### Haroon Pasha

An arm has a mass of 7.0 kg. Treating the arm as if it were a single point mass m attached to a rigid massless rod, determine the work that must be done by the deltoid muscle to move the arm from position 1 to position 2. (Position one and two have an angle of 30 degrees between them and it is 21.0 cm from the shoulder to the elbow of the arm)

By doing a little trigonometry, I figured out that the distance between position 1 and position 2 is about 0.12m. I will use the equation
W(for work)=F*delta x*cos(angle)

How do I find the force applied to move it from position one to two?

Haroon Pasha said:
An arm has a mass of 7.0 kg. Treating the arm as if it were a single point mass m attached to a rigid massless rod, determine the work that must be done by the deltoid muscle to move the arm from position 1 to position 2. (Position one and two have an angle of 30 degrees between them and it is 21.0 cm from the shoulder to the elbow of the arm)

By doing a little trigonometry, I figured out that the distance between position 1 and position 2 is about 0.12m. I will use the equation
W(for work)=F*delta x*cos(angle)

How do I find the force applied to move it from position one to two?

Your arm is in the Earth's gravitational field. You will have to do work against this force. The minimum work you have to do to move your arm 30° is the work done BY GRAVITY on your arm as it moves 30°. So let's find that. If we could find exactly what HEIGHT your arm had moved, it would be easy: W = -mgh. But we can't apparently. So we'll stick with the definition of work done by a force, which is the line integral

$$W = \int_{C}\vec{F}\cdot d\vec{r}$$

C is a portion of circle. Easy to parametrize. Let's do that..

$$x(\theta) = 0.21 sin(\theta)$$
$$y(\theta) = -0.21 cos(\theta)$$
$$0\leq \theta \leq 30°$$
$$\vec{r}(\theta) = 0.21 sin(\theta)\hat{x} -0.21 cos(\theta)\hat{y}$$
$$\vec{r'}(\theta) = 0.21 cos(\theta)\hat{x} + 0.21 sin(\theta)\hat{y}$$

$$\Rightarrow \int_{C}\vec{F}\cdot d\vec{r} = \int_0^{30°} \vec{F}(\vec{r}(\theta)) \cdot \vec{r'}(\theta)d\theta = \int_0^{30°}-0.21mgsin(\theta)d\theta = 0.21mg(cos(30)-1)$$

So your muscle will have to do work in the amount -0.21mg(cos(30)-1).

Last edited:
Haroon Pasha said:
An arm has a mass of 7.0 kg. Treating the arm as if it were a single point mass m attached to a rigid massless rod, determine the work that must be done by the deltoid muscle to move the arm from position 1 to position 2. (Position one and two have an angle of 30 degrees between them and it is 21.0 cm from the shoulder to the elbow of the arm)

By doing a little trigonometry, I figured out that the distance between position 1 and position 2 is about 0.12m. I will use the equation
W(for work)=F*delta x*cos(angle)

How do I find the force applied to move it from position one to two?
The force is mg. What is W in terms of mass, g, and the height?

AM

Wow. that makes sense. That was very helpful. Even though I am not in calc based physics, I still understoon your reasoning. Thanks Quasar987 and AM both.

It turns out we can get the height exactly directly from trigonometry. It would require a drawing but the key is

0.21 - h = 0.21 cos(30°)

## What is work?

Work is defined as the force applied to an object multiplied by the distance the object moves in the direction of the force. It is represented by the equation W = F * d.

## How is work calculated?

To calculate work, you must first determine the force applied to the object and the distance it moves in the direction of the force. Then, you can use the equation W = F * d to find the work done.

## What is the unit of measurement for work?

The unit of measurement for work is the joule (J). This unit is derived from the basic units of mass (kilograms), length (meters), and time (seconds).

## How does the angle of the force affect the work done?

The angle of the force affects the work done because work is only done when the force is applied in the same direction as the movement of the object. If the force is applied at an angle, only the component of the force in the direction of the movement will contribute to the work done.

## What factors can affect the work done to move an object from one position to another?

The work done to move an object from one position to another can be affected by various factors such as the magnitude and direction of the force applied, the distance the object moves, and the angle of the force. Other factors such as friction, air resistance, and the mass of the object can also affect the work done.