Question

A particle is located at coordinates $x=2.0\mathrm{m},y=3.0\mathrm{m}$ What is the magnitude of the torque about the origin when the particle is acted upon by a force of magnitude $5.0\mathrm{N}$ in $(a)$ the positive $x$ direction, $(b)$ the positive $y$ direction, and $(c)$ the negative $x$ direction?

Short answer

The magnitudes of the torque are 0 N.m along both the positive and negative x directions, and 5 sqrt(13) N.m along the positive y direction.

Step-by-step solution

Calculate the Position Vector's Magnitude

The magnitude of the position vector r = sqrt((2.0m)^2 + (3.0m)^2) = sqrt(4 + 9) = sqrt(13) m.

Compute the Torque for Different Force Directions

Now, for each force directions, the magnitude of the torque τ = |rF sinΘ| will be computed. Here, r = sqrt(13)m and F = 5.0N. For (a) the positive x direction, Θ is the angle with x axis which is 0. Hence, τ = rFsin0 = 0 N.m. For (b) the positive y direction, Θ is the angle with y axis which is 90°. Hence, τ = rFsin90 = sqrt(13)m . 5.0N = 5 sqrt(13) N.m. For (c) the negative x direction, Θ is the angle with negative x axis which is 180°. Hence, τ=rFsin180 = 0 N.m.

Conclusion

Therefore, the magnitude of the torque about the origin when the particle is acted upon by a force is 0 N.m along the positive x and negative x directions, and 5 sqrt(13) N.m along the positive y direction.

Key concepts

Position Vector

The position vector is a crucial concept in physics, particularly when dealing with problems involving torque. Simply put, a position vector is a straight line that extends from a fixed point (often the origin in coordinate systems) to the position of a particular object or particle.
In our exercise, the particle's position is given at coordinates $x=2.0\text{m},y=3.0\text{m}$. This means that the position vector, often denoted as $\mathbf{r}$, starts from the origin $(0,0)$ and ends at $2.0\text{m},3.0\text{m}$.

• The magnitude of the position vector $\mathbf{r}$ is calculated using the Pythagorean theorem. This is because the position vector forms the hypotenuse of a right triangle: \ $$|\mathbf{r}|=\sqrt{({2.0}^2+{3.0}^2)}=\sqrt{(4+9)}=\sqrt{13}\text{m}$$

This magnitude is essential because it represents the lever arm distance that plays a critical role in calculating torque. Without accurately defining the position vector and its magnitude, it would be impossible to proceed with further torque calculations.

Force Direction

In torque-related problems, the direction of the applied force is as crucial as the magnitude. The force's direction determines how much rotational effect it will have around a pivot point, like the origin.
In this exercise, we explore three distinct force directions:

• Positive $x$ direction
• Positive $y$ direction
• Negative $x$ direction

The direction of the force influences the angle formed between the position vector and the force vector, which in turn affects the torque magnitude calculated by $\tau =|\mathbf{r}||\mathbf{F}|\sin \mathrm{\Theta }$.

Let's remember:

• A force parallel to its line of action, like in the positive or negative $x$ directions, results in zero torque due to the $\sin (0)=0$ and $\sin (180)=0$.
• Force perpendicular to the line of action, like in the positive $y$ direction, maximizes torque, since the angle is ${90}^{\circ }$, and $\sin (90)=1$.

Understanding force direction in terms of angles helps to solve complex physics problems more intuitively.

Angle Measurement

Angles are measurements that assess the relationship between two intersecting lines or vectors at a vertex. In the context of torque, the angle $\mathrm{\Theta }$ between the position vector ($\mathbf{r}$) and force vector ($\mathbf{F}$) is a vital element.
Here's why angle measurement is key in our problem:

• The angle influences the value of the sine function $\sin \mathrm{\Theta }$, which affects the torque calculation.
• As seen in the exercise cases: - Positive $x$ direction: $\mathrm{\Theta }=0^{\circ }$ leads to zero torque. - Positive $y$ direction: $\mathrm{\Theta }={90}^{\circ }$ leads to maximum torque. - Negative $x$ direction: $\mathrm{\Theta }={180}^{\circ }$ also results in zero torque.

Understanding how to measure and interpret these angles allows you to predict the torque one might expect when applying forces in various directions relative to a pivot point. This intuitive grasp of angle measurement, along with the sin function, underpins many physics applications and everyday rotational mechanics.