Physics 163

Unit 3: Motion in Two-Dimensions

Problem Set E

Overview:

Problem Set E targets your ability to apply Newton's first and second laws of motion in order to analyze physical situations which involve forces which are directed at angles to the horizontal and vertical directions. This application will involve the use of vector principles and the trigonometric functions - sine, cosine and tangent. The following three types of physical situations will be analyzed:

• the hanging of signs and other objects by cables or wires which are stretched out at angles to the vertical (problems #1-#7)
• the motion of objects along both rough and smooth surfaces when a tension or applied force is exerted at an angle to the horizontal (problems #8-#13)
• the motion of objects along both rough and smooth inclined plane surfaces (problems #14-#20)

There are at least six skills and understandings which must be well understood to be successful on Problem Set E.

1. Newton's first law of motion

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 be at 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.

2. Newton's second law of motion

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 = Fnet / m . 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:

• If 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 vectorr 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.

3. Vector resolution

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. The diagram below illustrates these relationships.

4. 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 symbol g (sometimes referred to as the acceleration of gravity). Weight (or Fgrav) and mass are related by the equation: Fgrav = m • g.

5. Friction forces

An object which is moving (or event attempting to 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:

Ffrict = µ • Fnorm.

The symbol µ (pronounced muˆ) represents of the coefficient of friction and will be different for different surfaces.

The Hanging of Signs at Equilibrium

Problems #1-#7 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 #1-#7 will require that you make the connections discussed in the above paragraph. The unknown quantity tobe 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.

The Acceleration of Objects by Forces at Angles

Problems #8 - #13 will target your ability to analyze objects which are accelerated across horizontal surfaces by forces directed at angles to the horizontal. In Unit 2, Newton's second law was 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 (see above ). 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 the vertical component of the applied force (Fy). As such, the normal force plus the vertical component of the applied force is equal to the downward gravity force. That is,

Fnorm + Fy = Fgrav

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

Fnet = Fx - Ffrict = m • a

In problems #8-#13 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.

Inclined Plane Problems

Problems #14 - #20 will target your ability to analyze objects positioned on inclined planes, either accelerating along the incline or in equilibrium. As in all problems in this set, the analysis begins with the construction of a free-body diagram in which forces acting upon the object are drawn. This is shown below on the left. Note that the force of friction is directed parallel to the incline, the normal force is directed perpendicular to the incline, and the gravity force is neither parallel nor perpendicular to the incline. It is common practice in Fnet = m•a problems to analyze the forces acting upon an object in terms of those which are along the same axis of the acceleration and those which are perpendicular to it. On horizontal surfaces, we would look at all horizontal forces separate from those which are vertical. But on inclined surfaces, we would analyze the forces parallel to the incline (along the axis of acceleration) separate from those which are perpendicular to the incline. Since the force of gravity is neither parallel nor perpendicular to the inclined plane, it is imperative that it be resolved into two components of force which are directed parallel and perpendicular to the incline. This is shown on the diagram below in the middle. The formulas for determining the components of the gravity force parallel and perpendicular to the inclined plane are listed on the diagram. Once the components are found, the gravity force can be ignored since it has been substituted for by its components; this is illustrated in the diagram below on the right.

Once the gravity force has been resolved into its perpendicular and parallel components, the problem is approached like any Fnet = m•a problem. The net force is determined by adding all the forces as vectors. The two forces directed perpendicular to the incline balance each other and add to zero. One might write this as:

Fnorm = Fperpendicular

The net force is therefore the result of the two forces directed parallel to the incline. As always, the net force is the force in the direction of acceleration minus the force directed opposite of the acceleration. In this case,

Fnet = Fparallel - Ffrict

Once the net force is determined from the parallel component of gravity and the friction force, the acceleration can be determined from the ratio of net force to mass.

View Sample Problem Set.

 Problem Description Audio Link 1 Straight-forward calculation of the weight of a sign from its mass 2 Extension of problem #1; determination of the tension in a cable supporting a sign 3 Determination of the tension in a cable supporting a sign 4 Determination of the tension in a cable supporting a sign 5 Determination of the weight of a sign which is being supported by two cables 6 Determination of the mass of a sign which is being supported by two cables 7 Determination of the tension in a cable supporting a sign 8 Determination of the acceleration of an object being pulled across a friction-free surface by a force exerted at an angle to the horizontal 9 Determination of the acceleration of an object being pulled across a friction-free surface by a force exerted at an angle to the horizontal 10 Determination of the friction force acting upon an object being pulled at a constant velocity by a force exerted at an angle to the horizontal 11 Extension of problem #10; determination of the coefficient of friction 12 Determination of the acceleration of an object being pulled across a rough surface by a force exerted at an angle to the horizontal 13 Determination of the acceleration of an object being pulled across a rough surface by a force exerted at an angle to the horizontal 14 Determination of the acceleration of an object along a friction-free inclined plane 15 Determination of the friction force acting upon an object which slides down an inclined plane at a constant velocity 16 Extension of problem #15; determination of the coefficient of friction 17 Determination of the friction force acting upon an object which slides down an inclined plane at a constant velocity 18 Determination of the acceleration of an object along a inclined plane with a rough surface 19 Determination of the acceleration of an object along a inclined plane with a rough surface 20 Determination of the acceleration of an object along a inclined plane with a rough surface

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