Torque and angular acceleration

Rotational motion is the motion of objects around a center of rotation or a fixed point. This concept is central to understanding the physics of wheels, gears, and rotating machinery. Two fundamental concepts in rotational motion are torque and angular acceleration. These ideas are similar to force and linear acceleration in linear motion, but they apply specifically to rotation.

Torque:

Torque is a measure of how much a force applied to an object rotates that object. It's not just the amount of force, but also how far away from the axis of rotation the force is applied. You can think of torque as the rotational equivalent of linear force. The symbol for torque is the Greek letter tau (τ).

The formula for calculating torque is:

τ = r × F × sin(θ)

Where:

• τ is the torque.
• r is the distance from the axis of rotation to the point where the force is applied (lever arm).
• F is the magnitude of the applied force.
• θ is the angle between the force vector and the lever arm vector.

Imagine you are trying to open a door. If you push near the hinges, it takes more effort to open the door than if you push near the handle. This is because the lever arm (distance from the hinges) is longer when you push near the handle, which produces more torque.

As shown in the visual example, applying force to a door that is away from the hinges produces more torque, making it easier to open.

Angular acceleration

Angular acceleration is the rate of change of angular velocity. It tells us how quickly an object is speeding up or slowing down its rotation. The symbol for angular acceleration is the Greek letter alpha (α).

The formula for angular acceleration is:

α = Δω / Δt

Where:

• Δω is the change in angular velocity.
• Δt is the change in time.

Just like acceleration tells us how velocity changes in linear motion, angular acceleration tells us how angular velocity changes in rotational motion. For example, when you spin a top, it starts with a high angular velocity. As it spins, it gradually slows down due to friction. This change in angular velocity is explained by angular acceleration.

Relation between torque and angular acceleration

Torque and angular acceleration are closely connected through Newton's second law of rotation. This law states that the net torque acting on an object is equal to the product of its moment of inertia and angular acceleration. It can be written as:

τ = I × α

Where:

• τ is the net torque.
• I is the moment of inertia, a measure of an object's resistance to a change in its rotation.
• α is the angular acceleration.

Basically, given the same amount of torque, an object with a larger moment of inertia will accelerate slower than an object with a smaller moment of inertia.

For example, consider two wheels of the same size and shape, one made of rubber and the other of steel. When the same torque is applied to both, the steel wheel having the greater moment of inertia will have a lower angular acceleration than the rubber wheel.

The visual example above shows two wheels: a rubber wheel and a steel wheel. Applying the same torque to both results in a greater angular acceleration for the rubber wheel.

Examples to understand torque and angular acceleration

Example 1: Turning a wrench

A mechanic is using a wrench to turn a bolt. The mechanic applies a force of 50 N to the end of a 0.3 m long wrench. The angle between the force and the lever arm is 90 degrees, making the force and lever arm perpendicular.

Calculate the torque generated.

τ = r × F × sin(θ) = 0.3 , m × 50 , N × sin(90°) = 15 , Ncdot m

The torque produced is 15 N m. This torque helps in tightening or loosening the bolt.

Example 2: Bicycle wheel

A cyclist applies force to the pedals, which causes the bicycle wheel to rotate. Suppose the moment of inertia of the wheel is 0.5 kg·m², and the cyclist applies a torque of 10 N·m. What is the angular acceleration of the wheel?

α = τ / I = 10 , Ncdot m / 0.5 , kgcdot m^2 = 20 , rad/s^2

The wheel experiences an angular acceleration of 20 rad/s², which represents how quickly the cyclist increased the rotational speed of the wheel.

Example 3: Spinning disc

Imagine a disc is rotating on a table with a constant angular velocity. If you place your finger on the edge of the disc to stop it, you are applying a force against its motion and producing torque. This action changes the angular velocity of the disc, producing angular acceleration.

If the initial angular velocity of the disc is 10 radian/s and it stops in 5 seconds, what is the angular acceleration?

The change in angular velocity is:

Δω = 0 - 10 = -10 , rad/s

Thus, the angular acceleration is:

α = Δω / Δt = -10 , rad/s / 5 , s = -2 , rad/s^2

The negative sign indicates that the disk is slowing down.

Conclusion

Torque and angular acceleration are essential concepts in understanding rotational motion. Torque is similar to force in linear motion, while angular acceleration describes how quickly an object's rotational speed changes. By exploring real-world situations and using these principles, we better understand the rotational dynamics that govern everything from spinning wheels to orbital motion. Understanding these ideas not only enriches our knowledge of physics but also enhances our ability to apply these concepts to solve practical problems.

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