Week 3 Overview 
Last week, we covered multiple forces acting on an object. This week we will cover motion in two dimensions, inclined planes, circular motion, and rotation. 
Forces in Two Dimensions (1 of 2) 
So far you have dealt with single forces acting on a body or more than two forces that act parallel to each other. But in reallife situations, more than one force may act on a body. How are Newton’s laws applied to such cases? We will restrict the forces into two dimensions.
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Since force and acceleration are vectors, Newton’s law can be applied independently to the X and Yaxes of a coordinate system. For a given problem you can choose a suitable coordinate system. But once a coordinate system is chosen, we have to stick with it for that problem. The example that follows shows how to find the acceleration of a body when two forces act on it at right angles to each other. 
Forces in Two Dimensions (2 of 2) 
To find the resultant acceleration we draw an arrow OA of length 3 units along the Xaxis and then an arrow AB of length 4 units along the Yaxis. The resultant acceleration is the arrow OB with a length of 5 units. Therefore, the acceleration is 5 m/s2 in the direction of OB. Also when you measure the angle AOB with a protractor, we find it to be 53°.
The acceleration caused by the two forces is 5 m/s2 at an angle of 53°. 
Uniform Circular Motion 
When an object travels in a circular path at a constant speed, its motion is referred to as uniform circular motion, and the object is accelerated towards the center of the circle. If the radius of the circular path is r, the magnitude of this acceleration is ac = v2 / r, where v is its speed and ac is called the centripetal acceleration. A centripetal force is responsible for the centripetal acceleration, which constantly pulls the object towards the center of the circular path. There cannot be any circular motion without a centripetal force. 
Banking
When there is a sharp turn in the road or when a turn has to be taken at a high speed as in a racetrack, the outer part of the road or the track is raised from the inner part of the track. This is called banking. It provides an additional centripetal force to a turning vehicle so that it doesn’t skid.
The angle of banking is kept just right so that it provides all the centripetal force required and a motorist does not have to depend on the friction force at all.
Inclined Planes 
Forces on an Inclined Plane
The inclined plane is a device that reduces the force needed to lift objects. Consider the forces acting on a block on an inclined surface. The inclined surface exerts a normal force FN on the block that is perpendicular to the incline. The force of gravity, FG, points downward. If there is no friction, the net force, Fnet, acting on the block is the resultant of FN and FG. By Newton’s second law the net force must point down the incline because the block moves only along the incline and not perpendicular to it. The vector triangle shows that the magnitude of the net force is always less than the weight, FG. Example: A 2 kg block is on an incline which is only able to support 16 N of its weight. Find the acceleration of the block along the incline. Solution: First, from FG = mg, a 2 kg block weighs 20 N. Also, the normal force is given as 16 N. We now draw a free body diagram showing all the forces acting on the block: We can find the net force on the block using Pythagorean’s Theorem: 
Now, we can put Fnet into F=ma to find the acceleration of the block:
The block will accelerate down the incline at 6 m/s2.
Projectile Motion (1 of 2) 
When a ball is thrown or a shell is fired from a gun at an angle to the horizontal, the ball or the shell follows a curved path known as a parabola. The motion is in two dimensions and is called projectile motion. The moving object is called a projectile.
The vertical motion and horizontal motion can be analyzed separately for a projectile. The horizontal direction can be the xaxis and the vertical direction of the yaxis. 
Vertical Motion
In the vertical direction, only the force of gravity acts on the projectile. The vertical motion of the projectile is the same as the motion of an object thrown vertically upward with the same initial vertical velocity. If vy is the initial vertical component of velocity, the projectile will reach a maximum height of h = vy2 / (2 g), which is the maximum height reached by an object thrown vertically upward with velocity vy. The time for which the projectile remains in flight is again determined by the initial vertical component of velocity and is given by t = 2 vy / g, which is also the total time for which an object thrown vertically upward with velocity vy stays in the air. Using the navigation on the left, please proceed to the next page. 
Projectile Motion (2 of 2) 

Horizontal Motion
In the horizontal direction, there is no force acting on the projectile. So by Newton’s law, there is no acceleration in the horizontal direction. If the projectile is given an initial velocity of vx it remains the same throughout the duration of the flight. The horizontal distance, x, that the projectile travels is given by: x = vxt The total time of flight has already been stated to be 2 vy / g. Therefore, the range is: x = vx (2 vy / g) x = 2 Vx vy / g. The range is determined by both the horizontal and vertical components of velocity. 
Rotational Motion (1 of 2) 
Rotational Motion (2 of 2) 
Torque
When the doorknob is pushed or pulled, the force rotates the door about its hinge. The turning action depends on the product of the force component perpendicular to the lever arm and the lever arm. This quantity is called torque, τ, and its units are N m. Torque = Force perpendicular to lever arm x length of lever arm τ=Fperpr 
Newton’s Law for Rotation
Newton’s second law, F = ma, is for translation. For rotation, Newton’s law is τ= Iα.
In this equation
τ is the torque, I am the rotational inertia of the rigid body, and α its angular acceleration.
A pair of dumbbells that has its two masses farther away from each other, is much more difficult to twist than one in which the masses are close together. Rotational inertia, I, depends not only on the mass of the object but also on how this mass is distributed about the axis of rotation. The farther the mass is from the axis of rotation, the greater the rotational inertia.
Rotational Equilibrium 
An object is in rotational equilibrium when the net torque acting on the object is zero. This, in turn, means that the angular acceleration is zero. Rotational equilibrium defines when an object is in balance, and can also be used to define when a lever will be able to raise a mass. 
Week 3 Summary 
This week covered forces in two dimensions, circular motion, inclined planes, projectiles, and rotation. You learned how to apply Newton’s laws in situations with different types of motion. Relate the concepts covered in this week to more reallife situations. Get Business & Finance homework help today 
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