Consider a force F that acts on a particle with position vector r . If represents the torque of this force about the origin, which of the following is true?

Correct answer: C. r · = 0 and F · = 0 Torque is defined as the cross product of the position vector and the force vector: = r × F By the properties of the cross product, the resulting vector is always perpendicular to both r and F . Since the dot product of two perpendicular vectors is zero, we have: r · = r · ( r × F ) = 0 F · = F · ( r × F ) = 0 Thus, both dot products are zero.

Concepts Of Physics MCQ Edition [Volume 1]

Exam: Concepts Of Physics MCQ Edition [Volume 1]
Subject: Physics
Chapter: Rotational Mechanics

Concepts Of Physics MCQ Edition [Volume 1] · Physics · Rotational Mechanics

Consider a force F that acts on a particle with position vector r ....

Consider a force

\vec{F}

that acts on a particle with position vector

\vec{r}

. If

\vec{\Gamma}

represents the torque of this force about the origin, which of the following is true?

Options

A $\vec{r} \cdot \vec{\Gamma} \neq 0$ but $\vec{F} \cdot \vec{\Gamma} = 0$
B $\vec{r} \cdot \vec{\Gamma} \neq 0$ and $\vec{F} \cdot \vec{\Gamma} \neq 0$
C $\vec{r} \cdot \vec{\Gamma} = 0$ and $\vec{F} \cdot \vec{\Gamma} = 0$
D $\vec{r} \cdot \vec{\Gamma} = 0$ but $\vec{F} \cdot \vec{\Gamma} \neq 0$

Correct answer

C. $\vec{r} \cdot \vec{\Gamma} = 0$ and $\vec{F} \cdot \vec{\Gamma} = 0$

Step-by-step solution

Torque is defined as the cross product of the position vector and the force vector:

\vec{\Gamma} = \vec{r} \times \vec{F}

By the properties of the cross product, the resulting vector

\vec{\Gamma}

is always perpendicular to both

\vec{r}

and

\vec{F}

.

Since the dot product of two perpendicular vectors is zero, we have:

\vec{r} \cdot \vec{\Gamma} = \vec{r} \cdot (\vec{r} \times \vec{F}) = 0

\vec{F} \cdot \vec{\Gamma} = \vec{F} \cdot (\vec{r} \times \vec{F}) = 0

Thus, both dot products are zero.

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