Law of the Lever

A lever is one of the simple machines that has laid a foundation on engineering. It consists of two basic components: a beam fixed at a point called the fulcrum. The lever rotates on a point on itself, providing what we call leverage.

The Law of the Lever can be interpreted as such:

“As the lever rotates around the fulcrum, points farther from this pivot move faster than points closer to the pivot. Therefore, a force applied to a point farther from the pivot must be less than the force located at a point closer in, because power is the product of force and velocity (Uicker, Pennock and Shigley, 2010).”

Meaning to say, given two points a and b where a has a greater distance from the fulcrum, the input force applied in point a would be amplified, thereby increasing the output force expedited in point b. If the reverse was true, and the distance between the point of input force (point a) is shorter than the point of output force (point b), then the lever reduces the magnitude of the force.

For key takeaways, the further from the fulcrum a force is applied, then the less effort is needed to move the lever. 

Rotation Angle

When objects rotate about some axis—for example, when the compact disc rotates about its center—each point in the object follows a circular arc. Consider a line from the center of the CD to its edge. Each pit used to record sound along this line moves through the same angle in the same amount of time. The rotation angle is the amount of rotation and is analogous to linear distance. We define the rotation angle Δθ to be the ratio of the arc length to the radius of curvature:      

The arc length Δs is the distance traveled along a circular path. Note that r is the radius of curvature of the circular path.

We know that for one complete revolution, the arc length is the circumference of a circle of radius r. The circumference of a circle is 2πr. Thus for one complete revolution, the rotation angle is

The result is the basis for defining the units used to measure rotation angles, Δθ to be radians (rad), defined so that 2π rad = 1 revolution.

Angular Velocity

How fast is an object rotating? We define angular velocity ω as the rate of change of an angle. In symbols, this is 

where an angular rotation Δθ takes place in a time Δt. The greater the rotation angle in a given amount of time, the greater the angular velocity. The units for angular velocity are radians per second (rad/s).

Angular velocity ω is analogous to linear velocity v. To get the precise relationship between angular and linear velocity, we again consider a pit on the rotating CD. This pit moves an arc length in a time.

Rotational Inertia

Rotational inertia is the property of an object to counter the direction of its spin in relation to a rotational axis. This is a fundamental concept in physics that is used in problems involving angular momentum and how the distribution of mass correlates to the rotational motion or velocity of a spinning object. 

Rotational inertia is dependent on (1) the mass and (2) the distribution of the mass relative to the axis of rotation. Moreover, it is also important to account the momentum of the object. A mass that moves farther from the center or point of rotation is harder to change rotational velocity than a mass that is closer to the axis of rotation.

In essence, the heavier the mass of the rotating object, the harder it is to change the rotational velocity and direction of its spin. Keep in mind these handy lessens as they can help you better understand the concepts behind the challenge. We hope this helps, and good luck to all IRC participants! 

References:

– Uicker, John; Pennock, Gordon; Shigley, Joseph (2010). Theory of Machines and Mechanisms (4th ed.). Oxford University Press, USA. ISBN 978-0-19-537123-9.

– Lumen Learning , Physics

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A catapult is a ballistic device used to launch a projectile a great distance without the aid of gunpowder or other propellants.

A catapult uses the sudden release of stored potential energy to propel its payload.

Parts of a Catapult

• Bucket – a container use to hold the payload prior to release;
• Payload – the object that will be discharge in projectile motion;
• Arm – holds the Bucket that has a pivot connection at the base;
• Base and Frame – Supports the Catapult’s weight and action;
• Rope – stores potential energy by stretching or winding up while it is attached directly or indirectly to the arm;
• Restraining Rope – it serves as the trigger of the catapult once release;
• Counterweight – used in other type of catapult. Stores potential energy by setting it in a higher elevation and drop it once the restraining rope is released.

Energies involved in the catapult’s mechanism

There are three primary energy storage mechanism used in a catapult.

• Tension – is built by stretching the rope up to the maximum limit. When it is stretched, the potential energy stores in the rope, parallel to the direction of how it is stretched.
• Torsion -is built in the pivot point of the arm. the more you twist the rope, the greater energy you stored tangent to the center of rotation.
• Gravity – counterweight is one type of storing the potential energy by pulling a heavy object against the gravitational force.

Once the Payload is released in the Catapult, it will create a projectile motion towards the direction it is positioned.

Effect of Velocity in a Projectile Motion

Velocity is the distance travelled of an object over time. In a projectile motion, it affects the distance travelled of an object. The higher the velocity, the farther it can reach at the same angle.

To show you the effect, in image 2, we have a cannonball weighing 20kg that is shot at an angle of 45 degrees.

Effect of Angle in a Projectile Motion

Angle affects how far and how high the object will go in a projectile motion.

Using the simplified formula: R = (Vo2sin2Ø)/g

• R – Range
• Vo – Initial Velocity of the Object
• Ø – Angle of discharge in a projectile motion
• G – Gravitational force

We can compute for the maximum distance traveled.

Here are the samples images with same object and initial velocity but differs in the angle.

Do you see the changes in distance and height travelled by the cannonball at different angles?

At 45 Degrees, the vertical and horizontal forces are equal giving the cannonball the farthest distance it can reach at a given velocity and gravitational force.

While at 90 degrees, the cannonball can attain the maximum elevation it can reach at a same velocity and gravitational force

Effect of Gravity in a Projectile Motion

Every planet has its own gravitational pull. Here on earth, our gravitational pull is 9.807m/s², while on the moon it is 1.62m/s². This force is pulling the object towards the center of the planet. In a projectile motion, it affects the time an object will hit the ground, the distance it will travel, and the maximum height it can achieve.

In image No. 9, it shows changing the value of gravity in a 20kg cannonball that fires at 20m/s velocity at 45-degree angle.

The lower the gravitational pull, the farther the object can travel at a given instance.

You may also check https://phet.colorado.edu/sims/html/projectile-motion/latest/projectile-motion_en.html for you to explore more about projectile motion by varying the gravitational force and including Air resistance of the object.

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So what is the science, or more specifically the physics, behind soccer? You will find that forces is one of the principles that you have probably learned from school by now. But just to recap:

• Force – a push or a pull, acting on an object as a result of its interaction with another object. It is a vector quantity which means that it has both magnitude and direction. There are two broad categories of forces: contact force and action-at-a-distance force.

Contact force is the type that result when two interacting objects are perceived to be physically contacting with each other. Examples are the following: Frictional force, Tensional force, Normal force, Air resistance force and Applied forces.

• Frictional force – The force exerted by a surface as an object moves across it or makes an effort to move across it.
• Tensional force – the force that is transmitted through a string, rope, cable or wire when it is pulled tight by forces acting from opposite ends.
• Normal force – the support force exerted upon an object that is in contact with another stable object
• Air resistance force – a special type of frictional force that acts upon objects as they travel through the air.
• Applied force – force that is applied to an object by a person or another object.

Action-at-a-distance force, on the other hand, are those types of forces that result even when the two interacting objects are not in physical contact with each other, yet are able to exert a push or pull despite their physical separation. Examples are Gravitational force, Electrical force and Magnetic force.

• Gravitational force – the force with which the earth, moon, or other massively large object attracts another object towards itself.
• Electrical force – the repulsive or attractive interaction between any two charged bodies 
• Magnetic force – attraction or repulsion that arises between electrically charged particles because of their motion.

Now, let’s see what forces are applied on soccer ball.

When the soccer ball is at rest, the only forces acting upon it are the gravitational force and normal force which are equal and opposite in direction. Since the forces are balanced, the object remains at rest.

Following the Newton’s First Law of Motion, the Law of Inertia, the object will stay at rest or uniform in motion unless acted upon by an unbalanced force. In other words, to make the ball move initially, an applied force which is the kick should be applied. How hard the person kicks the ball will dictate the initial velocity and the angle of the trajectory (curved path).

Now, all this is explained in the situation in which we have gravity. In the first mission though, things are a bit different: the gravitational force is missing! Complete the first mission of Stage 1 and, based on your experience, try to write down how does an object behave when kicked in an environment where there is no gravity.

Source: https://www.physicsclassroom.com

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