Analytical Problems

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 real life situations more than one force may act on a body. How are Newton’s laws applied to such cases? We will restrict the forces to two dimensions.

Since force and acceleration are vectors, Newton’s law can be applied independently to the X and Y-axes 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)

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To find the resultant acceleration we draw an arrow OA of length 3 units along the X-axis and then an arrow AB of length 4 units along the Y-axis. The resultant acceleration is the arrow OB with the 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

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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 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

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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:

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Now, we can put Fnet into F=ma to find the acceleration of the block: image7.jpg

The block will accelerate down the incline at 6 m/s2.

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Projectile Motion (1 of 2)

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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 x-axis and the vertical direction the y-axis.

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)

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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)

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The CD drive in your computer makes the CD spin at a high rate to either read or write data. This is an example of rotation about a fixed axis. Until now, we have dealt with objects that translate or move along a straight or curved path.

Just as the motion along a path is described in terms of distance, speed, and acceleration, rotational motion is measured in terms of the angular position, speed, and acceleration of a body.

Angular Displacement, Velocity, and Acceleration

Angular displacement is the angle through which a body rotates and is measured in radians. When a body makes one full rotation its angular displacement, θ, is 360° or 2π radians or 6.28 radians. A radian is approximately 57.3°. You should always use radians when working with any circular motion problem. To convert between units, one revolution = 360 degrees = 2π radians.

To convert an angular displacement to a distance, multiply by the radius of the circle:

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A rotating rigid body has an angular speed. For translation, speed is the rate at which distance is covered. Angular velocity, ω, is the rate at which the angular position of the body changes.

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To convert from an angular velocity to a tangential velocity, multiply by the radius of the circle:

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If the angular velocity of a body changes, the body has an angular acceleration, α. Angular acceleration is the rate at which angular velocity changes.

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To convert from an angular acceleration to a tangential acceleration, multiply by the radius of the circle:

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Rotational Motion (2 of 2)

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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 the Newton’s law is τ= Iα.

In this equation

τ is the torque, I is 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

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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

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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 real life situations.