Force F⃗=(-8.0 N) î+(6.0 N) ĵ acts on a particle with position vector r⃗=(3.0 m) î+(4.0 m) j

Force F⃗=(-8.0 N) î+(6.0 N) ĵ acts on a particle with...

Force $\vec{F}=(-8.0 \mathrm{~N}) \hat{\mathrm{i}}+(6.0 \mathrm{~N}) \hat{\mathrm{j}}$ acts on a particle with position vector $\vec{r}=(3.0 \mathrm{~m}) \hat{\mathrm{i}}+(4.0 \mathrm{~m}) \mathrm{j} .$ What are (a) the torque on the particle about the origin and (b) the angle between the directions of $\vec{r}$ and $\vec{F} ?$

Force $\vec{F}=(-8.0 \mathrm{~N}) \hat{\mathrm{i}}+(6.0 \mathrm{~N}) \hat{\mathrm{j}}$ acts on a
particle with position vector $\vec{r}=(3.0 \mathrm{~m}) \hat{\mathrm{i}}+(4.0 \mathrm{~m}) \mathrm{j} .$ What are (a) the torque on the particle about the origin and (b) the angle between the directions of $\vec{r}$ and $\vec{F} ?$

Key Concepts

Torque

Torque is the measure of the rotational effect of a force applied to a particle or rigid body about a fixed point or axis. It is calculated as the cross product of the position vector (from the pivot point to the point of application of the force) and the force vector. This concept is central in rotational dynamics and helps in understanding how forces cause objects to rotate.

Vector Cross Product

The vector cross product is an operation on two vectors that results in a third vector perpendicular to the plane containing the original vectors. The magnitude of this vector is proportional to the product of the magnitudes of the two vectors and the sine of the angle between them. In the context of torque, the cross product provides both the magnitude and the direction of the torque vector.

Position Vector and Force Vector

Position vectors represent a point in space relative to a reference point, usually the origin, while force vectors represent the magnitude and direction of a force. Understanding these vectors and their components is essential in physics because many problems involve evaluating how forces affect motion and produce rotational effects.

Angle Between Vectors

The angle between two vectors quantifies the directional difference between them and is calculated using the dot product. In physics, knowing the angle between the force vector and the position vector is important because it influences the magnitude of the torque and the effective component of the force that contributes to rotation.

Dot Product

The dot product is a scalar product of two vectors, calculated as the product of their magnitudes and the cosine of the angle between them. It is particularly useful in determining the angle between vectors or the projection of one vector onto another, which is a fundamental step when computing the angle between the force direction and the position vector direction.

Transcript

00:01 So we've got this particle, and we are looking to find the torque around the origin for if it has a force of negative 8 newtons in the x direction and positive 6 newtons in the y direction applied to it when it's located at this point here.

00:17 So 3 meters in the x direction and 4 meters in the y direction.

00:22 And so we can find the torque two different ways.

00:27 The first way is r cross f.

00:32 Torque equals r cross f.

00:36 We can take that cross product in this way, where r is 3, 4, 0, and f is negative 8, 6 ,0.

00:51 And i'm abbreviating here.

00:54 For the x direction, we're going to get 4 times 0, 0 minus 0, so no x components.

01:01 For the y direction, we're going to get, again, 0 minus 0, so no y components.

01:08 And in the z direction, we're going to get 3 times 6 or 18 minus 4 times negative 8 or negative 32.

01:17 So 18 minus negative 32, we're going to get a positive 50 newton meters in the z direction.

01:29 That is our a.

01:31 So that's the result, that's the answer for part a, r cross f.

01:35 The other way we can do it is the cross product, i mean, is to take the magnitude of r times the magnitude of f and multiply it by the sign of the angle between them.

01:49 Well, part b asks what this angle is...

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