Angular Momentum & Newton's Second Law: Videos & Practice Problems

Angular momentum (L) relates to the rotational version of Newton's second law, where the sum of torques equals the change in angular momentum over time. The equation can be expressed as $\sum \tau =\frac{∆}{L}/∆t$. In a practical example, applying a torque of 80 N·m to a 10 kg object spinning at 180 RPM with a radius of 5 m results in a stopping time of approximately 59 seconds, illustrating the conservation of angular momentum and the effects of torque.

Angular Momentum & Newton's Second Law

Understanding the relationship between angular momentum and the rotational version of Newton's second law is crucial in physics. Newton's second law in its linear form is expressed as F = ma, which can also be represented in terms of linear momentum as ΣF = Δp/Δt, where Δp is the change in momentum over the change in time. The rotational equivalent of this law is given by Στ = Iα, where Στ represents the sum of torques, I is the moment of inertia, and α is the angular acceleration.

In the rotational context, we can rewrite the equation in terms of angular momentum L, leading to Στ = ΔL/Δt. Here, L is defined as L = Iω, where ω is the angular velocity. This relationship allows us to analyze problems involving rotating objects effectively.

For example, consider a point mass of 10 kg rotating at 180 RPM in a circular path with a radius of 5 meters. If a constant torque of 80 Nm is applied in the opposite direction to stop the object, we can determine the time it takes to come to a complete stop. The initial angular velocity ω can be calculated using the formula ω = 2πf, where f is the frequency in revolutions per second. Converting 180 RPM to radians per second gives us ω = 6π rad/s.

To find the moment of inertia I for a point mass, we use the formula I = m r², resulting in I = 10 kg × (5 m)² = 250 kg·m². Substituting these values into the equation Στ = ΔL/Δt allows us to solve for the time Δt it takes to stop the object:

Δt = (Iω_initial - Iω_final) / τ

Since the final angular velocity is 0 (the object comes to a stop), we have:

Δt = (250 kg·m² × 6π rad/s) / 80 Nm

Calculating this gives us approximately 59 seconds, indicating the time required for the object to stop under the applied torque. This example illustrates the practical application of angular momentum and torque in rotational dynamics, reinforcing the interconnectedness of these fundamental concepts in physics.

A solid disc of mass M = 40 kg and radius R = 2 m is free to rotate about a fixed, frictionless, perpendicular axis through its center. You apply a constant, tangential force on the disc's surface (as shown), to get it to spin. Calculate the magnitude of the force needed to get the disc to 100 rad/s in just one minute.

• A. 33.3 N
• B. 66.7 N
• C. 133 N
• D. 167 N

Answer: (B)

Here’s what students ask on this topic:

What is the relationship between angular momentum and Newton's second law?

The relationship between angular momentum (L) and Newton's second law in its rotational form is given by the equation:

$\sum \tau =\frac{∆L}{∆t}$

Here, the sum of torques (Στ) is equal to the change in angular momentum (ΔL) over the change in time (Δt). This is analogous to the linear version of Newton's second law, where the sum of forces equals the change in linear momentum over time. In essence, torque is to angular momentum what force is to linear momentum.

How do you calculate the stopping time of a rotating object using torque?

To calculate the stopping time of a rotating object using torque, you can use the equation:

$\sum \tau =\frac{∆L}{∆t}$

Rearrange to solve for Δt:

$∆t=\frac{∆L}{\sum \tau }$

First, find the initial angular momentum (L_initial) using L = Iω, where I is the moment of inertia and ω is the angular velocity. Then, calculate the change in angular momentum (ΔL = L_final - L_initial). Finally, divide ΔL by the applied torque (Στ) to find the stopping time.

What is the moment of inertia for a point mass?

The moment of inertia (I) for a point mass is calculated using the formula:

$I=m{r}^2$

Here, m is the mass of the object and r is the distance from the axis of rotation. This formula is crucial for solving problems involving rotational motion, as it allows you to determine the resistance of the object to changes in its rotational state.

How do you convert RPM to angular velocity?

To convert RPM (revolutions per minute) to angular velocity (ω) in radians per second, use the formula:

$\omega =\frac{2\pi \mathrm{RPM}}{60}$

Here, RPM is the rotational speed in revolutions per minute. Multiply RPM by 2π to convert to radians per minute, then divide by 60 to convert to radians per second. This conversion is essential for solving problems involving angular motion.

What is the significance of torque in rotational motion?

Torque (τ) is a measure of the rotational force applied to an object. It is significant in rotational motion because it determines the rate of change of angular momentum. The equation:

$\sum \tau =\frac{∆L}{∆t}$

shows that the sum of torques is equal to the change in angular momentum over time. This relationship is analogous to how force affects linear momentum in linear motion. Torque is crucial for understanding how rotational systems respond to applied forces.