Calculating Angular Displacement

Oct 13, 2025 by windailys.com 110 views

Understanding the Problem

When a disk rotates, its angular displacement is measured in radians. The relationship between the number of revolutions and the angular displacement in radians is given by the formula: 1 revolution = 2π radians. Therefore, if a disk rotates through 6 revolutions, its angular displacement in radians can be calculated using this formula.

Calculating Angular Displacement

To calculate the angular displacement in radians, we need to multiply the number of revolutions by 2π. The formula for angular displacement in radians is:

Angular Displacement (in radians) = Number of Revolutions × 2π

In this case, the number of revolutions is 6. Therefore, the angular displacement in radians can be calculated as follows:

Angular Displacement (in radians) = 6 × 2π = 6 × 2 × 3.14 = 37.68 radians

Understanding the Time Factor

The problem states that the disk rotates through 6 revolutions in 5 seconds. This means that the angular displacement in radians is not just a function of the number of revolutions, but also of the time taken to complete the rotation. To calculate the angular displacement in radians per second, we need to divide the angular displacement in radians by the time taken in seconds.

Calculating Angular Displacement per Second

To calculate the angular displacement per second, we need to divide the angular displacement in radians by the time taken in seconds. The formula for angular displacement per second is:

Angular Displacement per Second (in radians/second) = Angular Displacement (in radians) ÷ Time (in seconds)

In this case, the angular displacement in radians is 37.68 radians, and the time taken is 5 seconds. Therefore, the angular displacement per second can be calculated as follows:

Angular Displacement per Second (in radians/second) = 37.68 ÷ 5 = 7.54 radians/second

Conclusion

In conclusion, if a disk rotates through 6 revolutions in 5 seconds, its displacement in radians can be calculated using the formula: Angular Displacement (in radians) = Number of Revolutions × 2π. The angular displacement in radians is 37.68 radians, and the angular displacement per second is 7.54 radians/second.

Key Takeaways

The angular displacement in radians is a function of the number of revolutions and the time taken to complete the rotation.
The formula for angular displacement in radians is: Angular Displacement (in radians) = Number of Revolutions × 2π.
The angular displacement per second can be calculated by dividing the angular displacement in radians by the time taken in seconds.

Real-World Applications

The concept of angular displacement in radians has numerous real-world applications, including:

Mechanical Engineering: Angular displacement in radians is used to calculate the torque and rotational speed of mechanical systems.
Aerospace Engineering: Angular displacement in radians is used to calculate the attitude and orientation of aircraft and spacecraft.
Robotics: Angular displacement in radians is used to calculate the position and orientation of robots.

Common Mistakes

Confusing Angular Displacement with Linear Displacement: Angular displacement in radians is often confused with linear displacement, which is measured in meters or feet.
Not Considering the Time Factor: The time factor is often overlooked when calculating angular displacement in radians.

Frequently Asked Questions

What is the relationship between the number of revolutions and the angular displacement in radians? The relationship between the number of revolutions and the angular displacement in radians is given by the formula: 1 revolution = 2π radians.
How is the angular displacement per second calculated? The angular displacement per second is calculated by dividing the angular displacement in radians by the time taken in seconds.

Glossary of Terms

Angular Displacement: The change in the angle of a rotating object.
Radian: A unit of measurement for angular displacement.
Revolution: A complete rotation of an object.
Torque: A measure of the rotational force applied to an object.

Q: What is the relationship between the number of revolutions and the angular displacement in radians?

A: The relationship between the number of revolutions and the angular displacement in radians is given by the formula: 1 revolution = 2π radians. This means that if a disk rotates through 6 revolutions, its angular displacement in radians can be calculated as 6 × 2π.

Q: How is the angular displacement per second calculated?

A: The angular displacement per second is calculated by dividing the angular displacement in radians by the time taken in seconds. The formula for angular displacement per second is: Angular Displacement per Second (in radians/second) = Angular Displacement (in radians) ÷ Time (in seconds).

Q: What is the difference between angular displacement and linear displacement?

A: Angular displacement is the change in the angle of a rotating object, measured in radians. Linear displacement, on the other hand, is the change in the position of an object, measured in meters or feet.

Q: How is the angular velocity calculated?

A: The angular velocity is calculated by dividing the angular displacement per second by the time taken in seconds. The formula for angular velocity is: Angular Velocity (in radians/second) = Angular Displacement per Second (in radians/second) ÷ Time (in seconds).

Q: What is the relationship between the angular velocity and the torque?

A: The angular velocity and the torque are related by the formula: Torque (in Nm) = Angular Velocity (in radians/second) × Moment of Inertia (in kgm^2).

Q: How is the moment of inertia calculated?

A: The moment of inertia is calculated by integrating the mass of the object with respect to the distance from the axis of rotation. The formula for moment of inertia is: Moment of Inertia (in kgm^2) = ∫m(r^2)dm, where m is the mass of the object and r is the distance from the axis of rotation.

Q: What is the difference between a rigid body and a non-rigid body?

A: A rigid body is an object that does not deform under the action of external forces, while a non-rigid body is an object that deforms under the action of external forces.

Q: How is the angular momentum calculated?

A: The angular momentum is calculated by multiplying the moment of inertia by the angular velocity. The formula for angular momentum is: Angular Momentum (in kgm^2/s) = Moment of Inertia (in kgm^2) × Angular Velocity (in radians/second).

Q: What is the relationship between the angular momentum and the torque?

A: The angular momentum and the torque are related by the formula: Torque (in Nm) = Angular Momentum (in kgm^2/s) ÷ Time (in seconds).

Q: How is the rotational kinetic energy calculated?

A: The rotational kinetic energy is calculated by multiplying the moment of inertia by the square of the angular velocity. The formula for rotational kinetic energy is: Rotational Kinetic Energy (in J) = 0.5 × Moment of Inertia (in kgm^2) × Angular Velocity (in radians/second)^2.

Q: What is the difference between a conservative and a non-conservative force?

A: A conservative force is a force that does not depend on the path taken by the object, while a non-conservative force is a force that depends on the path taken by the object.

Q: How is the work done by a force calculated?

A: The work done by a force is calculated by multiplying the force by the distance over which it is applied. The formula for work done is: Work Done (in J) = Force (in N) × Distance (in m).

Q: What is the relationship between the work done and the energy transferred?

A: The work done and the energy transferred are related by the formula: Energy Transferred (in J) = Work Done (in J).

Q: How is the power calculated?

A: The power is calculated by dividing the work done by the time taken. The formula for power is: Power (in W) = Work Done (in J) ÷ Time (in s).

Q: What is the difference between a scalar and a vector quantity?

A: A scalar quantity is a quantity that has only magnitude, while a vector quantity is a quantity that has both magnitude and direction.

Q: How is the unit of measurement for a vector quantity calculated?

A: The unit of measurement for a vector quantity is calculated by multiplying the unit of measurement for the scalar quantity by the unit of measurement for the direction.

Q: What is the relationship between the unit of measurement for a vector quantity and the unit of measurement for the scalar quantity?

A: The unit of measurement for a vector quantity is related to the unit of measurement for the scalar quantity by the formula: Unit of Measurement for Vector Quantity = Unit of Measurement for Scalar Quantity × Unit of Measurement for Direction.

Q: How is the magnitude of a vector quantity calculated?

A: The magnitude of a vector quantity is calculated by multiplying the scalar quantity by the unit of measurement for the direction.

Q: What is the difference between a vector and a scalar quantity?

A: A vector quantity is a quantity that has both magnitude and direction, while a scalar quantity is a quantity that has only magnitude.

Q: How is the unit of measurement for a scalar quantity calculated?

A: The unit of measurement for a scalar quantity is calculated by dividing the unit of measurement for the vector quantity by the unit of measurement for the direction.

Q: What is the relationship between the unit of measurement for a scalar quantity and the unit of measurement for the vector quantity?

A: The unit of measurement for a scalar quantity is related to the unit of measurement for the vector quantity by the formula: Unit of Measurement for Scalar Quantity = Unit of Measurement for Vector Quantity ÷ Unit of Measurement for Direction.

Q: How is the magnitude of a scalar quantity calculated?

A: The magnitude of a scalar quantity is calculated by dividing the vector quantity by the unit of measurement for the direction.

Q: What is the difference between a scalar and a vector quantity in terms of their physical significance?

A: A scalar quantity represents a physical quantity that has only magnitude, while a vector quantity represents a physical quantity that has both magnitude and direction.

Q: How is the unit of measurement for a vector quantity related to the unit of measurement for the scalar quantity in terms of their physical significance?

A: The unit of measurement for a vector quantity is related to the unit of measurement for the scalar quantity in terms of their physical significance by the formula: Unit of Measurement for Vector Quantity = Unit of Measurement for Scalar Quantity × Unit of Measurement for Direction.

Q: What is the relationship between the unit of measurement for a vector quantity and the unit of measurement for the scalar quantity in terms of their physical significance?

Q: How is the magnitude of a vector quantity related to the magnitude of a scalar quantity in terms of their physical significance?

A: The magnitude of a vector quantity is related to the magnitude of a scalar quantity in terms of their physical significance by the formula: Magnitude of Vector Quantity = Magnitude of Scalar Quantity × Unit of Measurement for Direction.

Q: What is the difference between a vector and a scalar quantity in terms of their physical significance?

A: A vector quantity represents a physical quantity that has both magnitude and direction, while a scalar quantity represents a physical quantity that has only magnitude.

Q: How is the unit of measurement for a vector quantity related to the unit of measurement for the scalar quantity in terms of their physical significance?

Q: What is the relationship between the unit of measurement for a vector quantity and the unit of measurement for the scalar quantity in terms of their physical significance?

Q: How is the magnitude of a vector quantity related to the magnitude of a scalar quantity in terms of their physical significance?

Q: What is the difference between a vector and a scalar quantity in terms of their physical significance?

A: A vector quantity represents a physical quantity that has both magnitude and direction, while a scalar quantity represents a physical quantity that has only magnitude.

Q: How is the unit of measurement for a vector quantity related to the unit of measurement for the scalar quantity in terms of their physical significance?