2A1 Chapter 9 Angular Motion

What is angular displacement?

The angle through which a rotating body moves, measured in radians.

How is angular displacement calculated?

Angular displacement = Average angular velocity × time.

What is the unit of angular velocity?

Radians per second (rad/s).

What is the formula to convert revolutions per minute (rpm) to rad/s?

ω = (n × 2π rad/rev) / (60 s/min).

What does angular acceleration describe?

The change in angular velocity over a given time interval.

What is the conventional symbol for angular acceleration?

α (alpha).

How is angular acceleration calculated?

Angular acceleration = (Final angular velocity - Initial angular velocity) / Time interval.

What is the formula for calculating the moment of inertia for a rotating disk?

I = m × r², where m is mass and r is radius.

What is the radius of gyration?

An equivalent radius at which the mass of an object can be assumed to be concentrated.

What is torque?

A measure of the force that can cause an object to rotate about an axis.

What is the relationship between torque and moment of inertia?

Torque = Moment of inertia × Angular acceleration.

What is the formula for calculating kinetic energy of rotating masses?

Kinetic Energy = 1/2 × I × ω², where I is moment of inertia and ω is angular velocity.

Define brake power.

The power output of an engine, calculated from the work done over time.

What is mechanical efficiency?

The ratio of useful work output to total work input, expressed as a percentage.

How is power transmitted by a belt drive calculated?

Power = Torque × Angular velocity.

What is centrifugal force?

The apparent force that draws a rotating body away from the center of rotation.

What is centripetal force?

The force that acts on a body moving in a circular path, directed towards the center of rotation.

What is centripetal acceleration?

The acceleration directed towards the center of a circular path, calculated as v²/r, where v is linear velocity and r is radius.

How do you calculate the distance of movement of a governor due to centrifugal force?

This involves applying the principles of centrifugal force and the geometry of the governor mechanism.

What is the average angular velocity when a flywheel starts from rest and reaches a speed of 200 rpm in 60 seconds?

Average angular velocity = (0 + 20.94 rad/s) / 2 = 10.47 rad/s.

How do you convert angular displacement from radians to revolutions?

Revolutions = Angular displacement (rad) / (2π rad/rev).

What is the significance of the diameter of a pulley in angular motion calculations?

The diameter does not affect the answers for angular displacement, velocity, or acceleration, as these depend on angular measures.

What is the formula for calculating the average angular velocity during a run down?

Average angular velocity = (Initial angular velocity + Final angular velocity) / 2.

What is the angular acceleration if a flywheel rotating at 31.42 rad/s comes to rest in 10 seconds?

Angular acceleration = (0 - 31.42 rad/s) / 10 s = -3.142 rad/s².

What is the total angular displacement in radians for a flywheel that decelerates from 31.42 rad/s to rest in 10 seconds?

Angular displacement = (31.42 rad/s + 0) / 2 × 10 s = 157.1 rad.

What is the formula for the moment of inertia (I) of a flywheel?

I = mr², where m is mass and r is the radius.

How is the radius of gyration (k) related to the moment of inertia?

k = √(I/m), where I is the moment of inertia and m is mass.

What is the moment of inertia of a flywheel with a mass of 300 kg and a radius of 1 meter?

The moment of inertia is 300 kg·m².

What is the radius of gyration for a flywheel with a radius of 1 meter?

The radius of gyration is 0.707 m.

What is torque (T) in relation to force (F) and radius (r)?

T = F × r, where T is torque, F is force, and r is the radial distance.

What are the units of torque?

The units of torque are Newton meters (Nm).

What happens to torque if the force acts through the center of rotation?

The torque will be zero.

How is angular acceleration (α) related to linear acceleration (a)?

a = r × α, where r is the radius.

What is the formula for torque when mass is concentrated at the radius of gyration?

T = m × k × α, where k is the radius of gyration.

What is the kinetic energy (K) formula for rotating masses?

K = 1/2 × I × ω², where I is the moment of inertia and ω is angular velocity.

How do you convert rpm to rad/s?

ω = n × (2π/60), where n is the speed in rpm.

What is the formula for work (W) done by a rotating body for one revolution?

W = F × r × 2π.

What is the formula for power (P) related to torque (T) and angular velocity (ω)?

P = T × ω.

What is brake power (P) in terms of mass (m), gravity (g), and spring balance force (S)?

P_brake = (mg - S) × r × (2π/60).

What is the angular acceleration of a flywheel brought from standstill to 600 rpm in 40 seconds?

Angular acceleration α = 1.571 rad/s².

What is the kinetic energy of a flywheel with a mass of 500 kg and a radius of gyration of 0.5 m rotating at 150 rpm?

The kinetic energy is 15.425 kJ.

What is the relationship between torque and angular acceleration?

Torque is analogous to force, and angular acceleration is analogous to linear acceleration.

What is the formula for calculating torque when a force is applied at radius L?

T = F × L.

What is the significance of the radius of gyration in angular motion?

It helps in calculating the moment of inertia for bodies that are not concentrated at the rim.

How is the work done by a rotating body calculated for multiple revolutions?

W = T × θ, where θ is the angle in radians.

What is the angular velocity of a flywheel rotating at 300 rpm?

The angular velocity is 31.42 rad/s.

What does the term 'mechanical efficiency' refer to in the context of engines?

It refers to the ratio of useful work output to the total energy input.

What is the relationship between linear and angular versions of Newton's Second Law?

Torque is analogous to force, moment of inertia is analogous to mass.

What is the effect of applying a force in a direction other than tangential to the radius?

The tangential component must be determined to calculate torque.

How is the torque required calculated for a flywheel with a mass of 200 kg and radius of gyration of 500 mm?

Using T = m × k × α, where α is the angular acceleration.

What is the formula for brake power measured by the rope brake?

P_brake = (π n (mg * S - F)) / 60

What is the brake power produced when the radius is 600 mm, mass is 25 kg, and spring balance reads 50 N at 360 rpm?

4.42 kW

How is mechanical efficiency calculated?

η = (P_brake / P_indicated) * 100

If the indicated power is 7500 kW and brake power is 7000 kW, what is the mechanical efficiency?

93.3%

What is the formula for power transmitted by a belt drive?

P = (n (F1 - F2) r * π) / 60

What factors affect power transmission in a belt drive?

Belt tension, surface area of contact, and pulley speed.

What is the power transmitted by a pulley with a diameter of 300 mm, tension on the tight side 2000 N, and slack side 500 N at 200 rpm?

4.712 kW

How does belt thickness affect power transmission?

Increased belt thickness raises the effective radius, increasing power transmitted.

How can centripetal and centrifugal forces be calculated?

Using Newton's Second Law: F = m a, where a can be expressed as v^2/r or ω^2 r.

What is the formula for centripetal acceleration?

a_c = v^2 / r or a_c = ω^2 * r

What units are used for centripetal acceleration?

m/s²

What is the relationship between linear velocity and angular velocity?

v = r * ω

What happens to velocity in circular motion?

Velocity changes direction even if speed remains constant.

What is the effect of unbalanced forces on an object in motion?

It causes the object to change its state of motion.

What is the role of friction in power transmission by belt drives?

Friction prevents slipping and ensures the belt transmits power effectively.

What is the significance of the radius in power calculations for pulleys?

The radius affects the torque and power transmitted by the pulley.

What is the relationship between tension on the tight and slack sides of a belt?

The effective driving force is the difference between the two tensions.

What happens to the power transmitted if the radius of the pulley increases?

The power transmitted increases due to a larger effective radius.

What is the angular measurement unit for angular velocity?

Radians per second (rad/s)

What is the formula for calculating the change in velocity in circular motion?

Δv = r * θ

What is the relationship between angular acceleration and linear acceleration?

Angular acceleration is related to linear acceleration by the radius of the circular path.

What does Newton's first law of motion state?

An object in motion stays in motion unless acted upon by an unbalanced force.

What is the effect of a small angle on velocity vector calculations?

For small angles, sin(θ) ≈ θ and tan(θ) ≈ θ.

How does the tension in a belt affect the power transmitted?

Higher tension on the tight side increases the power transmitted.

What are the units of centripetal acceleration?

The units of centripetal acceleration are m/s².

What is the centripetal force formula?

Centripetal force (F_c) = m a_c = m (ω²r), where m is mass.

What is the centrifugal force on a rotating mass?

Centrifugal force (F_c) = m * ω²r, where m is mass, ω is angular velocity, and r is radius.

What is a governor in mechanical engineering?

A governor is a device used to control the speed of an engine, originally introduced by Isaac Watt.

How does a mechanical governor operate?

As speed increases, the governor's bobs move outward due to centrifugal force, adjusting the throttle valve.

What is the relationship between the height of a governor and speed?

The height of the governor is determined by the speed and is independent of the mass of the bob.

What is the formula for calculating the height of a governor?

Height (h) = (ω² * r) / g, where ω is angular velocity, r is radius, and g is acceleration due to gravity.

What happens if rotating masses are not balanced?

Unbalanced rotating masses create excessive vibration and can cause major damage.

How can you balance a single rotating mass?

By placing an equal mass at the opposite location and the same distance from the center of rotation.

What is the moment of force for a rotating mass?

The moment of force is calculated as m * r, where m is mass and r is the radius.

What is the angular velocity of a flywheel rotating at 200 rpm?

Angular velocity (ω) = (200 × π) / 30 = 20.94 rad/s.

What is the kinetic energy formula for a rotating mass?

Kinetic energy (KE) = 0.5 I ω², where I is the moment of inertia.

What is the mechanical efficiency of an engine lifting a mass?

Mechanical efficiency = (Output power / Input power) × 100%.

What is the difference between centripetal and centrifugal force?

Centripetal force acts towards the center of rotation, while centrifugal force appears to act outward on a mass in a rotating system.

How do you calculate the change in height of a Watt governor?

The change in height can be calculated based on the difference in angular speeds using the governor height formula.

What is the compensating mass required for balance?

The compensating mass is calculated based on the moments of the unbalanced masses.

What is the formula for calculating torque?

Torque (τ) = Force (F) × Radius (r), where F is the applied force and r is the radius.

What is the formula for calculating centrifugal force on a blade?

Centrifugal force (F_c) = m ω² r, where m is the mass of the blade, ω is angular velocity, and r is the distance from the center.

What is the importance of overspeed protection in rotating machinery?

Overspeed protection prevents mechanical failure by limiting the speed of rotating components.

How do you calculate the power transmitted by a pulley?

Power transmitted = (Tight side tension - Slack side tension) × Velocity of the belt.

What is the significance of the radius in balancing multiple rotating masses?

The radius affects the moment of force, which is crucial for achieving balance in multiple rotating masses.