Question

State, or find out, which of the following are scalars and which are vectors: (a) the volume of a petrol tank, (b) a length measured in metres, (c) a length measured in miles, (d) the angular velocity of a flywheel, (e) the relative velocity of two aircraft, (f) the work done by a force, (g) electrostatic potential, (h) the momentum of an atomic particle.

Short answer

Question: Identify whether the following quantities are scalars or vectors: (a) Volume of a petrol tank, (b) Length measured in metres, (c) Length measured in miles, (d) Angular velocity of a flywheel, (e) Relative velocity of two aircraft, (f) Work done by a force, (g) Electrostatic potential, (h) Momentum of an atomic particle. Answer: (a) Scalar, (b) Scalar, (c) Scalar, (d) Vector, (e) Vector, (f) Scalar, (g) Scalar, (h) Vector.

Step-by-step solution

(a) Volume of a petrol tank

The volume of a petrol tank only has a magnitude (measured in litres or gallons), and there is no associated direction. Thus, it is a scalar.

(b) Length measured in metres

A length measured in metres has a magnitude (the number of metres) but does not have a direction (since it's a pure distance, not a position). So, length is a scalar.

(c) Length measured in miles

Similar to (b), a length measured in miles has a magnitude, but no direction. Therefore, it is also a scalar.

(d) Angular velocity of a flywheel

Angular velocity has both a magnitude (the speed of rotation in radians per second or degrees per second) and a direction (the axis of rotation). Hence, it is a vector.

(e) Relative velocity of two aircraft

The relative velocity of two aircraft includes both a magnitude (speed difference) and a direction (the direction in which one aircraft is moving relative to the other). Therefore, it is a vector.

(f) Work done by a force

The work done by a force has only a magnitude (measured in joules) and no associated direction, as it is the product of force and the distance moved in the direction of the force. Thus, work done is a scalar.

(g) Electrostatic potential

Electrostatic potential has only a magnitude (measured in volts) and no associated direction. It is the amount of work needed to move a unit charge from one point to another in an electric field. Therefore, it is a scalar.

(h) Momentum of an atomic particle

The momentum of an atomic particle has both a magnitude (mass times velocity) and a direction (the direction of motion). Thus, momentum is a vector.

Key concepts

Angular Velocity

Angular velocity is a fascinating concept used in physics to describe how fast something spins around an axis. This is different from regular speed or velocity. Here, it’s all about rotation. The magnitude tells you how fast the object is spinning, usually measured in radians per second or degrees per second. But that’s not all! Angular velocity also includes the direction, which is crucial.

Imagine looking down at a spinning top. If it spins clockwise, the direction of its angular velocity is different than if it were spinning counterclockwise. It's a vector quantity, which means both direction and magnitude matter.

• Defined by both speed and direction of rotation
• Measured in radians or degrees per second
• Direction can be clockwise or counterclockwise

Recognizing angular velocity as a vector helps in many areas, from engineering to everyday situations, like noticing which way your watch hands move.

Relative Velocity

Relative velocity focuses on the motion of one object in relation to another. It's all about comparing speeds and directions between moving objects.

If two cars are traveling, the relative velocity helps you understand how fast one car is moving compared to the other. It's not just about the speed but the direction too. If two aircrafts are flying towards each other, their relative velocity is not only how fast they are approaching but also the direction in relation to one another.

Think of it like this: If both aircrafts are in-flight, and one is heading east while another is heading west, their relative velocity will take into account both their speeds and their opposing travels.

• Involves both magnitude (speed) and direction
• Relates two moving objects to each other
• Analyzes how one object moves about another

This concept helps in understanding motion better from different perspectives, making it a vector quantity too.

Momentum

Momentum is a key concept when discussing motion and force. It considers how much motion an object has and is calculated by multiplying an object's mass by its velocity.

Unlike just speed or velocity, momentum is a vector—it is defined by both the size of the momentum and the direction the object is moving toward. For example, if a soccer ball is kicked, its momentum depends on how heavy it is and how fast and in what direction it moves.

• Product of mass and velocity
• Includes both magnitude and direction
• Important for understanding forces and motion impacts

Knowing the momentum gives us valuable insights into how difficult it is to change the motion of objects, like stopping a rolling ball or a moving vehicle. Understanding momentum helps us analyze collisions and interactions between moving bodies.