Potential energy is always associated with interactions between two or more bodies; in the case of gravitational potential energy, we consider a body with mass in the gravitational pull of the Earth. GPE is the potential energy a body possesses based on its position relative to a reference level (usually the Earth’s surface). We can express GPE mathematically as:
where:
Notice that the higher a body is from the Earth's surface ground, the greater its GPE.
Example 2
Lift a box to a height of above the Earth’s surface. The gravitational potential energy of the box (relative to the Earth) is
Example 3
A tall stool sits next to you on the Earth’s surface. When you lift a box to a height of above the Earth's surface, the gravitational potential energy of the box (relative to the top of the stool) is now:
Work
Work is a term associated with our daily activities involving physical and mental stress, goals to accomplish, deadlines to meet, etc. Of course, the work we refer to as labor is different than the true definition of the word work (symbol W) we shall study in this chapter. When bodies need an applied force to move them, work is being done on the body by that applied force. The work that is done by the applied force causes the energy of the body to change. In this chapter we will study the work done by one or more forces on one or more bodies, determine the types of energy involved, and draw connections between the work done on the bodies and the energies changes in the bodies. First, however, we need to identify what we call work.
Identifying Work
Work, , is defined as the product
Where is the force applied to a body (either a push or a pull) and is the displacement of the body.
Using the units of are newtons, , and the units of “” are in “meters,” . So, work, , is in units of “Joules," where . Note that is never used as a unit of work (or energy); rather it is reserved as the unit for "torque."
The Free Body Diagram
To determine all the work being done on a single body, we need order to clarify all the forces acting on the body. To this end, a free-body diagram of that body is usually created. The purpose of the free-body diagram is twofold:
to treat the single body as a “point mass” having the same amount of mass as the original body, but with a volume concentrated at one point. The reason for doing this is to circumvent any rotation that may actually occur when one or more forces are applied to the body (you can’t rotate a “point”). We are only interested here in the work that causes the body to move in one dimension.
to show all the forces “on” the body of interest, not the forces the body may impose on other bodies.
In creating free-body diagrams, forces acting on the point particle are always drawn as pull forces. That is, the head of the vector arrow representing each force always points away from the point mass. Think about it. Pushing or pulling on a body in the same direction, with the same amount of force, creates the same motion of the body.
Work Done by a Single Force
Figure 8 above shows the work being done on a crate by applying a pull force of created by the man. The crate is initially at rest. Figure 8(a) shows a person pulling on the rope attached to the crate causing the crate to move a distance . We will assume that friction between the bottom surface of the crate and the floor is so small that we consider it negligible, so it will not affect the motion of the crate.
Figure 8(b) is a free-body diagram of the crate. The weight of the crate is balanced by the normal force on the crate due to the surface of the floor. The only force acting on the crate is the single force due to the tension force in the rope. As the person pulls on the rope, tension in the rope does work on the crate causing it to move a distance . The amount of work done on the crate by the tension force is
This is the maximum amount of work done on the crate by the rope.
Figure 9.12
A person pulling on a wooden crate that has a mass of using rope. We will consider the ropes mass to be so small that it can be neglected. The person pulls the crate with a force of magnitude , for a distance . There is between the box and the floor. The net effect of all these forces is the movement of the box to the left. (b) A free-body diagram of the crate.
Work Done by Several Forces
If more than one force is applied to a body, the net or total work, , on a body is the sum of the individual works done by the individual forces.
Example 4
The wooden crate in Figure 8 is acted upon by two forces in the vertical direction, gravity (pulling downward) and the normal force (pushing upward) from the ground.
The total work, , done by these two forces on the wooden crate is
Because the two forces are balanced, the box does not move in the vertical direction. Hence, . Note that the work done by the normal force is always zero.
The Work—Energy Theorem
Work causes bodies to change their energy. The total work, , done by all forces acting on a body changes where m is the mass of the object and v is the speed of the object. This can be expressed mathematically as
Because the unit of KE is joules, the unit of is also joules. The person in Figure 8 increases the kinetic energy of the crate by due to the total work done on the crate.
In Figure 9, a person applies a pushing force, to wooden crate of mass crate. There is no friction between the crate and the floor due to the wheels underneath the crate. The increase in kinetic energy due to the work done by over a distance causes the crate to increase its initial velocity from , to a final velocity of . That is
Solving for , we get .
Figure 9.13
A person pushing a crate , initially at rest , with a constant applied force, , on ground. The direction of is indicated by the red arrow. The crate is pushed a distance, d, causing work, W, to be done on it by the . The work increases the velocity of the crate from to .
Work of Two Applied Forces on an Object
Consider the situation shown below in Figure 10. Person and dog are each pulling on a crate that has a mass of . Both person and the dog are pulling on a rope attached to the crate. We will consider the rope so light that we can neglect its mass. There is friction between the crate and the floor.
Figure 9.14
Person 1 and dog 2 are each pulling on a crate using ropes (with negligible mass) attached to the crate. The crate has a mass of . Person pulls with a force of magnitude . The dog pulls with a force of magnitude . The magnitude of the kinetic frictional force between the crate and the floor is .
In what follows, answer the questions pertaining to Figure 10.
Procedural Steps
In the box to the right of the dog in Figure 10, draw a free-body diagram for the crate, showing all the forces on it. Label each force and its magnitude.
The crate in Figure 10 moves a distance to the right. Calculate the work done by each applied force on the crate and the total work, , done on the crate, , , and
Determine the kinetic energy change, , of the crate.
Determine the final velocity of the crate.
Work and GPE
Work done by gravity on a body can also cause a change in the gravitational potential energy, GPE, of the body. Figure 11 shows a crate, originally at height, h, from the ground, acted upon by gravity. Because gravity is a conservative force, the work done by gravity alone on a body is independent of the path taken by the body. In this case, an equal amount of work is done on the crate by gravity alone when the crate goes from height, h, to ground level either down a frictionless hill, or down a series of steps.
The work done by gravity, , is
The change in GPE for the crate going from height, , to ground is
Therefore
Figure 9.15
The brown crate has two paths, and , available to it to descend from height, , to ground level . Path is a ramp with negligible friction inclined at to the horizontal (ground). Path is a series of steps. The work done by gravity
, , to the crate from height to ground level is the same for each path. is independent of the path taken and only depends on the difference in height, . Free-body diagrams of crate for paths and are shown in the small boxes above the crates.
Application of Work to Various Bodies
The Mass-Pulley System
One type of system consisting of more than one body is the mass-pulley system. Figure 12 consists of two identical crates ( and ) made up of identical masses and connected by a massless string that runs over a frictionless pulley. There is tension, , in the rope. Crate is initially held at rest on a frictionless surface.
At time , crate is released and the two crates move a distance of upon which crate hits the pulley and stops. Note: A motion sensor, placed on the left side of crate, detects the instantaneous velocity of crate , can be used to determine the kinetic energy, KE, of each crate.
Figure 9.16
The mass-pulley system is made up of two identical masses (crate and crate ) connected by a string that runs over a pulley with negligible friction. Crate is initially held at rest on table top. The friction between crate and the table top can also be considered negligible. Both masses are . When crate is released, both crates move .
Procedural Steps
In the boxes to the right of the mass-pulley system in Figure 12, draw a free-body diagram for each crate, showing all the forces on it. Label each force and its magnitude.
Determine the work done by each force on crate and crate in Figure 12 and the total work done on both crates: , , .
Determine the change in kinetic energy, , of each crate as a result of the movement.
Determine the change in gravitational potential energy, , of each crate as a result of the movement.
The Inclined Plane
The inclined plane, a simple machine devised by ancient man, is used even today as a routine way to help move bodies more easily to a higher level. The sloping surfaces at loading docks and the wheelchair ramps outside hospitals and schools are just a few applications of the inclined plane. In this section, we will apply several ideas of forces and work involving a frictional force to the inclined plane.
It takes more effort and force to lift a body up a staircase or ladder than to simply push a body up a frictionless inclined plane. The inclined plane therefore has a mechanical advantage. When comparing the length of a staircase or ladder to the length of an inclined plane going from the same height, , to ground level, the length of the staircase or ladder is much shorter. The trade-off for putting in less effort and force in pushing a body up an incline is increased displacement. As we explained earlier, the amount of work done by gravity to lower a body form a certain height, , to ground level is independent of the path because gravity is a conservative force. The amount of work done against gravity to raise a body from ground level to height, , is, therefore, also independent of the path.
Figure 13 shows a crate of mass, , placed on a ramp inclined at an angle of to the horizontal ground. The coefficients of static friction and kinetic friction are and , respectively, between the crate and the incline. The straight-line distance from the crate to the end of the incline is . Note: A motion sensor, placed at the bottom of the incline, detects the instantaneous velocity of crate , which can be used to determine the kinetic energy, KE, of the crate.
Figure 9.17
This illustrates a crate placed on a ramp that is inclined to the horizontal. The coefficients of static friction and kinetic friction are and , respectively, between the crate and the incline. The straight-line distance from the crate to the end of the incline is .
Procedural Steps
In the boxes to the right of the mass-pulley system in Figure 13, draw a free-body diagram of the crate, showing all the forces on it. Label each force and its magnitude.
Determine the work done by each force on the crate and the total work, , done on the crate.
Using the work-energy theorem, determine the change in kinetic energy, , of the crate as a result of the movement.
Determine the change in gravitational potential energy, , of the crate as a result of the movement down the incline. Is gravity a “conservative” force? (i.e., work done is independent of the path taken)
Review Questions
Starting from ground level, a pulling force, pulls a crate for up a ramp at constant speed. The ramp makes an angle of with the horizontal ground. The vertical rise is . 1.Calculate the work done by alone, .
2.Calculate work done by gravity, .
3.Calculate the work done by kinetic friction, , in pulling the crate up the inclined plane.
4.Calculate the of crate when it reaches the vertical height.
Virginia Physics Standards of Learning
This chapter fulfills sections PH.1, PH.2, and PH.5 of the Virginia Physics Curriculum.
Chapter 10: Laboratory Activities
Bruce Davidson. "Laboratory Activities", 21st Century Physics Flexbook.
Purpose
The purpose of this chapter is to provide several examples of physics experiments that utilize century technology. The technology highlighted in this chapter is the PASCO GLX Handheld Interface. The Xplorer GLX captures, analyzes, stores, and prints data quickly without the use of a computer. It can also be connected to a computer to make use of the datastudio graphing software.
Most of the included labs are intended to be used by teachers that are new to this technology. As a teacher becomes comfortable with the technology, more advanced and inquiry based labs are easily done as extensions.
The following labs are written specifically for the GLX. However, they can be easily modified to use the PASCO PasPort Interface for collecting data with a computer. Older analog Interfaces from PASCO, such as the 500 and 750 Science Workshop Interface can use PASPORT digital and analog adapters that allow you to use PASCO's latest technology to be used without having to replace sensors.
Although the PASCO technology is documented in this chapter, other companies such as Data Harvest Educational Inc., Fourier Systems Inc., Texas Instruments Inc., and Vernier Software & Technology Inc., also offer probeware technology.
It is hoped that this chapter, which is only a beginning, will spark interest with physics teachers to begin using century technology. If teachers are already using probeware technology, then this may serve as an additional resource. The idea of the physics FlexBook is an evolving supplemental physics resource. Additions to this chapter might include generic laboratory experiments that can be followed with any brand of probeware.
Lab #1 Position—Match Graph Lab (PASCO GLX)
Figure 10.1
Position-Match Graph Lab
Lab Description
The purpose of this activity is to explore graphs of motion (position versus time). The activity uses a motion sensor to measure the motion as a student moves back and forth in front of a flat reflector along a straight line at different speeds. Download Lab: http://www.pasco.com/file_downloads/experiments_of_month/glx/position_match_with_glx/Position-Match-with-GLX.zip.
Lab #2 Velocity of a Motorized Cart (PASCO GLX)
Figure 10.2
Velocity of a Motorized Cart
Lab Description
This activity uses a motion sensor to measure the motion of a motorized cart as it moves at different speeds. Although constant velocity is straightforward, the graphical representation of constant velocity involves many fundamental concepts of kinematics. The slope of a plot of position versus time is the speed of the object. Students will describe the relationship between the slope for each plot of data and the physical quantities represented by the slope. Download Lab: http://www.pasco.com/file_downloads/experiments/pdf-files/glx/physics/03-Vel-of-cart-SV.pdf.
Lab #3 Acceleration Due to Gravity (PASCO GLX)
Figure 10.3
Acceleration Due to Gravity
Lab Description
This activity uses the motion sensor to measure the motion of a ball as it falls and bounces. The motion of the ball is reco
rded and displayed, allowing students to analyze the position and velocity of the ball. A velocity versus time graph can be used to find the acceleration of the ball. Students will compare the experimental value of acceleration (slope of velocity versus time) to the accepted value for the acceleration due to gravity. Download Lab: http://www.dentonisd.org/512719301176/lib/512719301176/_files/05_Free_fall_SV.pdf
Lab #4 Acceleration on an Inclined Track (PASCO GLX)
Figure 10.4
Acceleration on an Inclined Track
Lab Description
This activity uses a motion sensor to measure the motion of a cart as it moves up and down an inclined plane. The Xplorer GLX is used to record and display the motion. From the collected data, students can determine whether the acceleration up and down the inclined plane is constant. Download Lab: http://www.aug.edu/hbusch/Phsc1011%20Files/Lab%202%20Accel%20on%20an%20inclined%20track.pdf.
Lab #5 Newton’s First Law —No Net Force (PASCO GLX)
Lab Description
This activity uses the motion sensor and PASCO GLX to measure the motion of a cart as it experiences different applied forces while traveling along a track. The purpose of this activity is to investigate the motion of an object when there is no net force applied compared to the motion when there is a net force applied. Download Lab: http://www.bayhicoach.com/pdfs/III%20Newtons%20First%20Law-No%20Net%20Force%20Activity.pdf
CK-12 21st Century Physics: A Compilation of Contemporary and Emerging Technologies Page 21