Video: Finding the Angular Momentum of a Particle and the Torque on It

A particle of mass 5.0 kg has position vector 𝐫 = (2.0𝐒 βˆ’ 3.0𝐣) m at a particular instant of time, 𝑑. At the instant 𝑑, the velocity of the particle with respect to the origin is 𝐯 = (3.0𝐒) m/s. What is the angular momentum of the particle at the instant 𝑑? If a force 𝐅 = (5.0𝐣) N acts on the particle at the instant 𝑑, what is the torque about the origin in that instant?

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Video Transcript

A particle of mass 5.0 kilograms has position vector 𝐫 equals 2.0𝐒 minus 3.0𝐣 meters at a particular instant of time 𝑑. At the instant 𝑑, the velocity of the particle with respect to the origin is 𝐯 equals 3.0𝐒 meters per second. What is the angular momentum of the particle at the instant 𝑑? If a force 𝐅 equals 5.0𝐣 newtons acts on the particle at the instant 𝑑, what is the torque about the origin in that instant?

Let’s highlight the vital information given in this two-part problem. We’re told that the mass of the particle we’re considering is 5.0 kilograms and that it has a position vector 𝐫 equals 2.0𝐒 minus 3.0𝐣 meters. The particle is moving with a velocity of 𝑣 equals 3.0𝐒 meters per second. And in part two, we’re told that the particle was subjected to a force 𝐅 equals 5.0𝐣 newtons.

In part one, we want to solve for the angular momentum of the particle at time 𝑑; we’ll call that capital 𝐋. And in part two, we’re asked to solve for the torque about the origin; we’ll represent that using the Greek letter 𝛕.

Both angular momentum and 𝛕, the torque, are vectors, meaning they have both magnitude and direction. Let’s begin our solution by summarizing the information given and recalling the vector relationship for angular momentum.

As we work towards solving for the angular momentum of the particle, let’s recall the relationship between 𝐫 and 𝐯 for angular momentum. The angular momentum of an object is equal to the position vector of that object crossed with the object’s momentum.

We know we can write momentum 𝑝 as the object’s mass times its velocity, so angular momentum equals 𝐫 cross π‘š times 𝐯. As we apply this equation to our scenario, we can factor out the π‘š so that the equation reads 𝐋 equals π‘š times the quantity 𝐫 cross 𝐯.

Now it will be important to recall mechanically how do we compute the cross-product between two vectors. In this case, we have 𝐫 and 𝐯 given to us as part of the problem statement. To know how to combine these in the cross-product, let’s recall the general equation for the cross-product between two three-dimensional vectors 𝐀 and 𝐁.

The cross-product between two vectors is equal to their components combined according to this equation. Notice that the resulting vector has an 𝐒-, a 𝐣-, and a 𝐀-component and that each one of these components is comprised of the components of 𝐀 and 𝐁 that are perpendicular to that direction. For example, the 𝐒th component of 𝐀 cross 𝐁 involves the 𝑦- and 𝑧-components of 𝐀 and 𝐁, respectively.

Let’s apply this cross-product equation to the given information in this problem. First, let’s figure out the 𝐒th component of the angular momentum 𝐋.

When we write out the 𝐒-, 𝐣-, and 𝐀-components of the angular momentum, we see, as we look at the first 𝐒th component, that that component is equal to π‘Ÿ sub 𝑦 times 𝑣 sub 𝑧 minus π‘Ÿ sub 𝑧 times 𝑣 sub 𝑦.

But for the position and velocity vectors given, there is no 𝑧-component, so 𝑣 sub 𝑧 and π‘Ÿ sub 𝑧 are both zero. We have both those terms being zero; the entire 𝐒th component is zero as well.

So now we move on to the 𝐣th component of our angular momentum. Again, we see π‘Ÿ sub 𝑧 in the first term and 𝑣 sub 𝑧 in the second. Since those are both still zero, the 𝐣th component of 𝐋 is zero too.

Now on to the last component, the 𝐀-direction component: π‘Ÿ sub π‘₯ minus 𝑣 sub 𝑦 minus π‘Ÿ sub 𝑦 times 𝑣 sub π‘₯. Looking at our velocity vector, it has only an 𝑖- or π‘₯-direction component. That means 𝑣 sub 𝑦 is zero.

π‘Ÿ sub 𝑦 is negative 3.0 and 𝑣 sub π‘₯ is 3.0. So our 𝐀-component simplifies to negative negative 3.0 times 3.0, or 9.0. We can now rewrite our expression for angular momentum in a simplified way. 𝐋, the angular momentum of our system, is equal to the mass π‘š multiplied by 9.0 meters squared per second in the 𝐀-direction.

We’re given π‘š as 5.0 kilograms. And when we multiply these two numbers together, we find that the overall angular momentum 𝐋, the product of these two numbers, is 45 kilograms meters squared per second in the 𝐀-direction. That’s the magnitude and direction of the system’s angular momentum.

In part two of our problem, we’re told that a force 𝐅 equals 5.0 newtons in the 𝐣-direction acts on the particle. And we want to solve for the torque this creates.

Let’s recall the vector relationship for torque. The torque that acts on an object is equal to the position vector of that object cross the force. In our problem, we have both 𝐫 and 𝐅, and we need to take a cross-product between them to solve for torque 𝛕.

Let’s write out the general equation for the cross-product between 𝐫 and 𝐅 using our cross-product formula. With the components of torque written out, we can now evaluate the 𝐒-, 𝐣-, and 𝐀-components of 𝛕 to see what their value is.

Looking at the 𝐒th component, we see that involves 𝐹 sub 𝑧 in the first term and π‘Ÿ sub 𝑧 in the second term. Looking at our 𝐫 and 𝐅 vectors, we see that neither of them has a 𝑧-component, so 𝐹 sub 𝑧 and π‘Ÿ sub 𝑧 are both zero. That means that the 𝐒-component of 𝛕 is zero as well.

Moving on to the 𝐣-component, we again see π‘Ÿ sub 𝑧 in the first term and 𝐹 sub 𝑧 now in the second term. These values are zero and so so is the 𝐣th component of 𝛕.

Moving to the last component, the 𝐀th component, π‘Ÿ sub π‘₯ is 2.0, 𝐹 sub 𝑦 is 5.0, π‘Ÿ sub 𝑦 is negative 3.0, and 𝐹 sub π‘₯ is zero. This means the 𝐀th component of 𝛕 can be written as 2.0 times 5.0.

When we multiply these numbers together, we get 10. So this tells us the overall torque acting on our particle is simply 10 newton meters acting in the 𝐀-direction. We’ve now solved for both the angular momentum and the torque that acts on this particle.

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