Rotational Motion

RotationalMotion ASUPAN, Nhanina FELICIO, Carmelle LICONG, Cindy LISONDRA, Denise CANANGCA-AN, Eric TINIO, Dime

Angles in Radians In science, angles are often measured in radians (rad). When the arc length s is equal to the radius r, the angle θ swept by r is equal to 1 rad. Any angle θ measured in radians is defined as,

Convert 110° into radians.

Angular Displacement In translational kinematics, the position of the body is defined as the displacement from a certain reference point. In rotational kinematics, the position of a point on a rotating body is defined by the angular displacement from some reference line that connects this point to the axis of rotation.

The body has rotated through the angular displacement if the point which was originally at P1 is now at the point P2. This angular displacement is a vector that is perpendicular to the plane of the motion. The magnitude of this angular displacement is the angle θ itself.

If it is positive, the rotation of the body is counterclockwise and the angular displacement vector points upward. If it is negative, the rotation is clockwise and the vector points downward.

which means that

Given: r = 1.25 m = 2.25 m Find: A boy rides on a merry-go-round at a distance of 1.25 m from the center. If the boy moves through an arc length of 2.25 m, through what angular displacement does he move? Solution:

Angular Velocity It is denoted by the lowercase of the Greek letter omega (ω) and is defined as the ratio of the angular displacement to the time interval, the time it takes an object to undergo that displacement. In the limit that the time interval approaches zero, becomes the instantaneous speed, v.

Angular velocity is expressed in: • radians per second (rad/s) • revolutions per second (rps) • revolutions per minute (rpm) 1 rev = 2π rad

Linear velocity of a point on the rotating body and angular velocity of the body are linked by the equation, s = rθ divided by t. but we know that and And so, The farther the distance r that the body is from the axis of rotation, the greater is its linear or tangential velocity.

A merry-go-round is rotating at a constant angular velocity of 5.4 rad/s. What is the frequency of the merry-go-round in revolutions per minute? Given: Find: Solution: 1 rev = 6.28 rad/s

Angular Acceleration Angular acceleration occurs when angular velocity changes with time. It is denoted by the symbol alpha, α.

There is a connection between the instantaneous tangential acceleration (linear motion) and angular acceleration (rotational motion).

Given: A figure skater begins spinning counterclockwise at an angular speed of 5.0π rad/s. She slowly pulls her arms inward and finally spins at 9.0π rad/s for 3.0 s. What is her average angular acceleration during this time interval? Find: Solution: 2

Torque To make an object start rotating, a force is needed; the position and direction of the force matter as well. The perpendicular distance from the axis of the rotation to the line along which the force acts is called the lever arm.

Here, the lever arm for FA is the distance from the knob to the hinge; the lever arm for FD is zero; and the lever arm for FC is as shown. The torque is defined as:

Rotational Inertia An object rotating about an axis tends to continue rotating about that axis unless an unbalanced external torque (the quantity measuring how effectively a force causes rotation) tries to stop it.

The resistance of an object to changes in its rotational motion is called rotational inertia which is also termed as moment of inertia. Torque is required to change the rotational state of motion of an object. If there is no net torque, a rotating object continues to rotate at a constant velocity.

Rotational inertia depends on the distribution of the mass. A large mass which is at a greater distance from the axis of rotation, has a greater moment of inertia than a large mass which is near the axis of rotation. The larger the moment of inertia of a body, the more difficult it is to put that body into rotational motion or, the larger the moment of inertia of a body, the more difficult it is to stop its rotational motion.

The moment of inertia of a single mass m, rotating about an axis, a distance r from m, we have I = mr2 The unit for the moment of inertia is kg•m2 and has no special name.

Find the moment of inertia of a solid cylinder of mass 3.0 kg and radius 0.50 m, which is free to rotate about an axis through its center. Given: Find: Solution: m = 3.0 kg r = 0.50 m I = ½mr2 = ½(3.0 kg)(0.50 m)2 = ½(3.0 kg)(0.25 m2) I = 0.38 kg • m2 Find: I

First law for rotational motion: A body in motion at a constant angular velocity will continue in motion at that same angular velocity, unless acted upon by some unbalanced external torque. Second law for rotational motion: When an unbalance external torque acts on a body with moment of inertia I, it gives that body an angular acceleration α, which is directly proportional to the torque τ and inversely to the moment of inertia. Third law of rotational motion: If body A and body B have the same axis of rotation, and if body A exerts a torque on body B, then body B exerts an equal but opposite torque on body A.

Angular Momentum If the rotational equivalent of force is torque, which is the moment of the force, the rotational equivalent of linear momentum (p) is angular momentum (L), which is the moment of momentum. Product of the moment of inertia of a rotating body and its angular velocity Unit is kg•m2/s

If an object is small compared with the radial distance to its axis of rotation, the angular momentum is equal to the magnitude of its linear momentum mv, multiplied by the radial distance r.

What is the angular momentum of a 250-g stone being whirled by a slingshot at a tangential velocity of 6 m/s, if the length of the slingshot is 30 cm? Given: m=250 g=0.25kg v = 6 m/s r = 30 cm=0.30 m Find: Solution: 2

Conservation of Angular Momentum Law of conservation of angular momentum states that: in the absence of an unbalanced external torque, the angular momentum of a system remains constant.

Key Concepts • 1. The angle turned through by a body about a given axis is called angular displacement. • 2. Angular velocity is the change in the angular displacement of a rotating body about the axis of rotation with time. • 3. Angular acceleration is the change in the angular velocity of a rotating body with time. • 4. The moment of inertia / rotational inertia is the measure of the resistance of a body to change in its rotational motion. • 5. The larger the moment of inertia of a body, the more difficult it is to change its rotational motion. • 6. Angular momentum is the product of the moment of inertia of a rotating body and its angular velocity. • 7. The law of conservation of angular momentum states that, if the total external torque acting on a system is zero, then there is no change in the angular momentum of the system.

Rotational Motion