**Unit 8: Newton's Laws
Applications**

**Problem Set D**

**Overview:**

Problem Set D targets your ability to combine trigonometric principles with Newton's second law in order to analyze physical situations involving forces exerted upon objects at angle to the horizontal and vertical axes. There are at least six skills and understandings which must be well understood to be successful on Problem Set D.

If an object is either at rest or in motion with a constant velocity, then that object is not accelerating. The lack of acceleration indicates that all the forces acting upon the object are balanced. That is to say, the net force upon the object is 0 Newtons. Such an object is said to beat equilibrium. The analysis of such objects at equilibrium will involve the understanding that the sum of all upward forces or force components equal the sum of all downward forces or force components. Similarly, the sum of all rightward forces or force components equal the sum of all leftward forces or force components.

The acceleration of objects is caused by an unbalanced or net force. The magnitude of the acceleration is equal to the ratio of net force to mass:a = F. Problems involving acceleration will often ask you to determine the net force, the mass or the magnitude of individual forces acting upon an object. There are typically two types of these problems:_{net}/ mIf the acceleration of an object is known, then the magnitude of the net force can usually be determined. This net force value is related to the vector sum of all individual force values; as such, the magnitude of an individual force can often be found if the net force can be calculated.

If the values of all individual force values are known, then the net force can be calculated as the vector sum of all the forces. The mass is often stated or determined from the weight of the object. The acceleration of the object can then be found as the ratio of the net force to mass.

A force which is directed at an angle to the horizontal and vertical axes can be resolved or broken into two parts - one which is horizontal and the other which is vertical. These horizontal and vertical parts are referred to as the components of the vector. The process of resolving a vector into its components requires the use of the sine, cosine and tangent functions. The force directed at an angle can be drawn as the hypotenuse of a right triangle. The direction of the force is often expressed as an angle with the horizontal or vertical; this angle is equivalent to one of the angles inside of the right triangle. The components of the force vector simply correspond to the vertical and horizontal sides of the right triangle. The sine and cosine functions can be used to determine the magnitude of the force components.

**Mass-weight relationship**

Mass is a quantity which is dependent upon the amount of matter present within an object; it is measured in kilograms and is independent of location. Weight, on the other hand, is the force of gravity which acts upon an object. Since gravitational forces vary with location, the weight of an object on the Earth's surface is different than its weight on the moon. Being a force, weight is expressed in the metric unit as Newtons. Every location in the universe is characterized by a gravitational constant represented by the symbolg(sometimes referred to as the acceleration of gravity). Weight (or F_{grav}) and mass are related by the equation:F._{grav}= m • g

An object which is moving (or eventattemptingto move) across a surface encounters a force of friction. Friction force results from the two surfaces being pressed together closely, causing intermolecular attractive forces between molecules of different surfaces. As such, friction depends upon the nature of the two surfaces and upon the degree to which they are pressed together. The friction force can be calculated using the equation:F._{frict}= µ • F_{norm}The symbol µ (pronounced

muˆ) represents of the coefficient of friction and will be different for different surfaces.

There are two main categories of problems in this set: 1) analysis of the accelerated or constant velocity motion of an object acted upon by a force exerted at an angle to the horizontal, and 2) the analysis of objects which are hung at equilibrium by two or more forces acting upward at an angle.

**The Acceleration of
Objects by Forces at Angles**

Problems #1 - #12 will target your ability to analyze objects which are moving across horizontal surfaces by forces directed at angles to the horizontal. Previously, Newton's second law has been applied to analyze objects accelerated across horizontal surfaces by horizontal forces. When the applied force is at an angle to the horizontal, the approach is very similar. The first task involves the construction of a free-body diagram and the resolution of the angled force into horizontal and vertical components. Once done, the problem becomes like a Unit 2 problem in which all forces are directed either horizontally or vertically.

The free-body diagram above shows the presence of a friction force. This force may or may not be present in the problems you solve. If present, its value is related to the normal force and the coefficient of friction (see above). There is a slight complication related to the normal force. As always, an object which is not accelerating in the vertical direction must be experiencing a balance of all vertical forces. That is, the sum of all up forces is equal to the sum of all down forces. But now there are two up forces - the normal force and As such, the normal force plus the vertical component of the applied force is equal to the downward gravity force. That is,

There are instances in which the applied force is exerted at an
angle below the horizontal. Once resolved into its components, there
are two downward forces acting upon the object - the gravity force
and the the vertical component of the applied force (F_{y}).
In such instances, the gravity force plus the vertical component of
the applied force is equal to the upward normal force. That is,

As always, the net force is the vector sum of all the forces. In
this case, the vertical forces sum to zero; the remaining horizontal
forces will sum together to equal the net force. Since the friction
force is leftward (in the *negative* direction), the vector sum
equation can be written as

In problems #1-#12 of Set E, you will need to use trigonometric functions to determine the components of the applied force. If friction is present, then you will need to determine the normal force in order to determine the friction force value. Then the net force can be computed using the above equation. And the acceleration can be found using Newton's second law.

**The Hanging of Signs at
Equilibrium**

Problems #13-#20 will target your ability to analyze objects suspended at equilibrium by two or more wires, cables, or strings. In each problem, the object is attached by a wire, cable or string which makes an angle to the horizontal. As such, there are two or more tension forces which have both a horizontal and a vertical components. The horizontal and vertical components of these tension forces is related to the angle and the tension force value by a trigonometric function (see above). Since the object is at equilibrium, the vector sum of all horizontal force components must add to zero and the vector sum of all vertical force components must add to zero. In the case of the vertical analysis, there is typically one downward force - the force of gravity - which is related to the mass of the object. There are two or more upward force components which are the result of the tension forces. The sum of these upward force components is equal to the downward force of gravity.

Problems #13-#20 will require that you make the connections discussed in the above paragraph. The unknown quantity to be solved for could be the tension, the weight or the mass of the object; the angle is always known. The graphic above illustrates the relationship between these quantities. Detailed information and examples of equilibrium problems is available online at The Physics Classroom.

**Additional Readings/Study
Aids:**

The following pages from The Physics Classroom tutorial may serve to be useful in assisting you in the understanding of the concepts and mathematics associated with these problems.

Determining Acceleration from Force | Determining Individual Force Values | Friction Force

Vector Components | Vector Resolution | Equilibrium | Net Force Problems Revisited

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