http://msuperl.org/wikis/pcubed/ 2026-05-03T14:55:46+00:00 http://msuperl.org/wikis/pcubed/ http://msuperl.org/wikis/pcubed/lib/tpl/bootstrap3/images/favicon.ico text/html 2014-11-20T16:28:53+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:a_meter_stick_on_the_ice&rev=1416500933&do=diff Example: A Meter Stick on the Ice Consider a meter stick whose mass is 300 grams and that lies on ice (see representation in which we are looking down on the meter stick). You pull at one end of the meter stick, at right angles to the stick, with a force of 6 newtons. Assume that friction with the ice negligible. What is the rate of change of the center-of-mass speed $v_{CM}$$\omega$$v_{CM}$$\omega$$\frac{d\vec{P}}{dt}$$\vec{F}_{net}$$\frac{d\vec{L}_{rot}}{dt}$$\vec{\tau}_{net,CM}$$\tau = r_{A}… text/html 2026-04-07T18:54:34+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:a_ping-pong_ball_hits_a_stationary_bowling_ball_head-on&rev=1775588074&do=diff Example: A Ping-Pong Ball Hits a Stationary Bowling Ball Head-on In an orbiting spacecraft a Ping-Pong ball of mass m (object 1) traveling in the +x direction with initial momentum $\vec{p}_{1i}$ hits a stationary bowling ball of mass M (object 2) head on, as shown in the figure in representations. $\vec{p}_{1i}$$\vec{p}_{1f} + \vec{p}_{2f} = \vec{p}_{1i} + \vec{p}_{2i}$$K = \frac{1}{2}(\frac{p^{2}}{m})$$$\vec{p}_{1f} + \vec{p}_{2f} = \vec{p}_{1i} + \vec{p}_{2i}$$$\vec{p}_{1f} \approx -\vec{p}_… text/html 2014-10-31T21:39:26+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:a_rebounding_block&rev=1414791566&do=diff Example: A Rebounding Block A metal block of mass 3 kg is moving downward with speed 2 m/s when the bottom of the block is 0.8 m above the floor. When the bottom of the block is 0.4m above the floor, it strikes the top of a relaxed vertical spring 0.4 m in length. The stiffness of the spring is 2000 N/m.$v_{i}$$v_{i}$$v_{f} = 0$$U_{g} \approx mgy$${\Delta K_{Earth}}$$E_{f} = E_{i} + W$$K = \dfrac{1}{2}mv^2$$U_{spring} = \dfrac{1}{2}k_{s}s^2$$U_{gravitational} = mgy$$E_{f} = E_{i} + W$$K_{f} + … text/html 2014-11-06T02:32:03+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:a_rod_rotating_not_around_its_center&rev=1415241123&do=diff Example: A Rod Rotating Not Around Its Center A thin rod with mass 140 grams and 60 cm long rotates at an angular speed of 25 radians/s about an axle that is 20cm from one end of the rod (see Figure in Representations). What is its kinetic energy? Facts $K_{tot} = K_{trans} + K_{rot} = \frac{1}{2}(Mr^{2}_{CM} + I_{CM})\omega^{2}$$K_{trans} = \frac{1}{2}Mr^{2}_{CM}$$K_{rot} = \frac{1}{2}I_{CM}\omega^{2}$$I = (\frac{1}{12})ML^{2}$$(r_{CM} = 0.1 m):$$K_{tot} = K_{trans} + K_{rot}$$K_{tot} = \frac… text/html 2014-11-06T02:59:56+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:a_yo-yo&rev=1415242796&do=diff Example: A Yo-yo You're playing with a yo-yo of mass m on a low-mass string (See Diagram in Representations). You pull up on the string with a force of magnitude F, and your hand moves up a distance d. During this time the mass falls a distance h (and some of the string reels off the yo-yo's axle). $m$$\Delta K_{trans}$$\int_i^f \vec{F}_{net} \cdot d\vec{r}_{cm}$$\Delta E_{sys}$$W_{surr}$$K_{trans}$$\Delta K_{trans}$$\int_i^f \vec{F}_{net} \cdot d\vec{r}_{cm}$$F_{net}$$d\vec{r}_{cm}$$\Delta K_{… text/html 2014-10-28T13:59:50+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:an_electric_heater&rev=1414504790&do=diff Example: An Electric Heater An electric heater receives an energy input of 5000 J of electric energy. During this time the heating element is maintained at a constant temperature. What is ${\Delta E_{thermal}}$ for the heater, and what is Q, the energy transfer between the heater's hot heating element and the cooler air?${\Delta E_{thermal}}$${\Delta E_{sys}} = W + Q$${\Delta E_{thermal}} = {\Delta E_{sys}} = 0 $$E_{sys}$${\Delta E_{sys}} = W + Q$$0 = Q + 5000 J$$Q = -5000 J$ text/html 2014-11-20T16:30:31+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:angular_momentum_of_halley_s_comet&rev=1416501031&do=diff Example: Angular Momentum of Halley's Comet The highly elliptical orbit of Halley's comet is shown in the representations. When the comet is closest to the Sun, at the location specified by the position vector $\vec{r}_{1}$ (“perihelion”), it is $8.77$ x $10^{10}$m from the Sun, and its speed is $5.46$$10^4$$\vec{r}_{2}$$1.32$$10^{4}$$1.19$$10^{12}$$\theta$$17.81^{\circ}$$2.2$$10^{14}$$\vec{r}_{1}$$ x $$\vec{r}_{1}$$5.46$$10^4$$\vec{r}_{2}$$1.32$$10^{4}$$\vec{r}_{2}$$1.19$$10^{12}$$\theta$$17.8… text/html 2014-07-10T20:23:08+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:averagevelcompare&rev=1405023788&do=diff Example: Comparing two ways of calculating the average velocity You have learned about two ways of computing the average velocity. The arithmetic average is an approximation and it can be a poor one. Consider the driving from East Lansing to Chicago (222 miles or 358 km). To get to Chicago, you drive at 55.0 mph (24.6 $\dfrac{m}{s}$$\dfrac{m}{s}$$\times10^5m$$\dfrac{m}{s}$$\dfrac{m}{s}$$v_{avg,x} = \dfrac{\Delta x}{\Delta t}$$v_{avg,x} \approx \dfrac{v_i + v_f}{2}$$$v_{avg,x} = \dfrac{\Delta x}… text/html 2018-02-09T18:33:14+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:calcgravforce&rev=1518201194&do=diff Example: Calculating the gravitational force exerted by the Earth on the Moon. At a particular moment in time the Moon is located $\langle 1.9\times10^8, 0, -1.9\times10^8 \rangle$ m in a coordinate system in which the origin is located at the center of the Earth. Determine the gravitational force exerted by the Earth on the Moon.$\langle 1.9\times10^8, 0, -1.9\times10^8 \rangle$$\langle 0,0,0 \rangle$$ 6.7\times10^{-11} Nm^2/kg^2 $$5.9\times10^{24} kg$$7.3\times10^{22} kg$$ \vec{F}_{grav} = -… text/html 2014-07-22T04:55:08+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:calculating_the_force_due_to_a_stretched_spring&rev=1406004908&do=diff Example: Calculating the force due to a stretched spring A spring with a mass block at the end of it and with a stiffness of 8 $N/m$ and a relaxed length of 20 $cm$ is attached to a chamber wall that results in its oscillations being horizontal. At a particular time the location of the block mass is $\langle .38,0,0 \rangle\,m$$L_0=0.2\,m$$8 N/m$$k_s=8\,N/m$$\vec{L} = \langle .38,0,0 \rangle m$$\langle 0,0,0 \rangle\,m$$ {\vec F_{spring}} = -k_s\vec{s}$$ |\vec{s}| = |L - L_0|$$$ {\vec F_{sprin… text/html 2014-10-01T05:24:54+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:deer_slug_example&rev=1412141094&do=diff Example: Firing a deer slug. How much force does a 12 gauge exert on your shoulder when firing a deer slug? Facts Mass of gun = 3.5kg Mass of slug = 0.22kg Lacking $\vec{F}_{net}$ on shoulder Approximations & Assumptions ${\Delta t} \longrightarrow 1/24s$ - Based on when a gun is fired in a movie, it usually occurs at about one movie frame, therefore, the collision time is less than 1/24s.$\vec{V}_{Slug} \longrightarrow 500m/s$$\vec{F}_{net} = \dfrac{\Delta\vec{p}}{\Delta t}$$\vec{p}_{s… text/html 2014-11-20T16:30:59+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:earth_s_translational_angular_momentum&rev=1416501059&do=diff Example: Earth's Translational Angular Momentum Calculate the magnitude of the Earth's translational (orbital) angular momentum relative to the Sun when the Earth is at location A and when the Earth is at location B as displayed in the representation of the situation in the representations. The mass of the Earth is $6$$10^{24}kg$$1.5$$10^{11}m$$6$$10^{24}$$1.5$$10^{11}$$2\pi r$$\vec{p} = m\vec{v}$$v = s/t$$\left|\vec{L}_{trans}\right| = \left|\vec{r}_A\right|\left|\vec{p}\right|\sin \theta$$2\p… text/html 2026-04-07T18:50:48+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:elastic_collision_of_two_identical_carts&rev=1775587848&do=diff Example: Elastic Collision of Two Identical Carts Cart 1 collides with stationary cart 2, which is identical. Suppose that the collision is (nearly) elastic, as it will be if the carts repel each other magnetically or interact through soft springs. In this case there is no change of internal energy. What are the final momenta of the two carts?$\vec{p}_f = \vec{p}_i + \vec{F}_{net} \Delta t$$E_f = E_i + W + Q$$K = \frac{1}{2}mv^{2} = \frac{1}{2}m(\frac{p}{m})^{2} = \frac{1}{2}(\frac{p^{2}}{m})$$… text/html 2014-10-28T13:07:32+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:energy_in_a_spring-mass_system&rev=1414501652&do=diff Example: Energy in a Spring-Mass System A mass of 0.2 kg is attached to a horizontal spring whose stiffness is 12 N/m. Friction is negligible. At t = 0 the spring has a stretch of 3 cm and the mass has a speed of 0.5 m/s. a: What is the amplitude (maximum stretch) of the oscillation?$s_{i}$$v_{i}$$v_{f}$$s_{i}$$v_{i}$$s_{f}$$E_{f} = E_{i} + W$$K = \dfrac{1}{2}mv^2$$U_{spring} = \dfrac{1}{2}k_{s}s^2$$E_{f} = E_{i} + W$$K_{f} + U_{f} = K_{i} + U_{i} + W$$K_{f}$$W$$0 + \dfrac{1}{2}k_{s}s^2_{f} =… text/html 2014-07-14T17:08:18+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:finalloccf&rev=1405357698&do=diff Example: Predicting the location of an object undergoing constant force motion The fan cart in the video below is observed to accelerate uniformly to the right. The air exerts a constant force on the blades that is around $0.45 N$. Determine the how far the fan cart has traveled after $2.2 s$$0.45 N$$2.2 s$$\dfrac{m}{s^2}$$m_{cart} = 0.3 kg$$\vec{F}_{net} = \sum \vec{F}_i = \langle 0.45, 0, 0 \rangle N$$x$$x_{f} - x_{i} = v_{xi} \Delta t + \dfrac{1}{2}\dfrac{F_{net,x}}{m} \Delta t^2$$\Delta x_{… text/html 2014-07-11T13:20:01+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:finalp&rev=1405084801&do=diff Example: Predicting the final momentum & velocity using the Momentum Principle Henrik Zetterberg is passing a hockey puck at a Red Wings practice. From video of the pass, you can determine the stick was in contact with the puck for $0.05 s$. You estimate the force with which $100 N$$m_{puck}=0.17kg$$0.05 s$$0.05 s$$100 N$$\vec{p}_f = \vec{p}_i + \vec{F}_{net} \Delta t$$$\vec{p}_f = m_{puck}\vec{v}_f = \vec{F}_{net} \Delta t$$$$\vec{v}_f = \dfrac{\vec{F}_{net}}{m_{puck}} \Delta t$$$$v_f = \dfrac… text/html 2015-09-17T12:16:15+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:finding_the_range_of_projectile&rev=1442492175&do=diff Example: Finding the range of a projectile In the previous example of time of flight, the out of control bus is forced to jump from a location $\langle 0,40,-5 \rangle$m with an initial velocity of $\langle 80,7,-5 \rangle m/s^{-1}$. We have now found the time of flight to be 3.65s and now want to find the position of where the bus returns to the ground. $\langle 0,40,-5 \rangle$$\langle 80,7,-5 \rangle$$\dfrac{m}{s^2}$$$ \vec{r}_f = \vec{r}_i + \vec{v}_{avg} \Delta t $$$$ x_f = x_i + V_{avg,x}… text/html 2015-09-15T16:51:13+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:finding_the_time_of_flight_of_a_projectile&rev=1442335873&do=diff Example: Finding the time of flight of a projectile An out of control bus is forced to jump from a location $\langle 0,40,-5 \rangle$m with an initial velocity of $\langle 80,7,-5 \rangle$m. Determine the time at which the bus hits the ground. Facts * Starting position of the bus $\langle 0,40,-5 \rangle$m * Initial velocity of the bus $\langle 80,7,-5 \rangle$$\dfrac{m}{s^2}$$$\vec{r}_{f} = \vec{r}_{i} + \vec{v}_{i} \Delta t + \dfrac{1}{2}\dfrac{\vec{F}_{net}}{m} \Delta t^2$$$\Delta{t}$… text/html 2014-09-22T04:16:59+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:holding_block_against_a_wall&rev=1411359419&do=diff Example: Holding a Block Against a Wall. You hold a 3 kg metal block against a wall by applying a horizontal force of 40 N, as shown in the figure in representations. The coefficient of friction for the metal-wall pair of materials is 0.6 for both static friction and sliding friction. Does the block slip down the wall?$\vec{F}_{net}$$\vec{F}_{net}$$\Delta \vec{p} = \vec{F}_{net} \Delta t$$ x: \Delta p_x = (F_{hand} - F_N) \Delta t = 0 $$ y: \Delta p_y = (F_N - mg) \Delta t, \,\,assuming\, it\, … text/html 2014-07-18T06:05:23+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:impulse&rev=1405663523&do=diff Example: Calculating the change in momentum In another example, we determined the net force acting on the fan cart in the video below. Now, you want to determine the change in momentum of the fan cart over a 4.2 second interval. Setup You need to compute the momentum change of the fan cart using the information provided and any information that you can collect or assume.$\dfrac{m}{s^2}$$\vec{F}_{cart} = \langle 0.45, 0, 0 \rangle N$$m_{cart} = 0.3 kg$$F_{air} = 0.45 N$$\vec{F}_{net} = \sum … text/html 2014-07-20T23:03:18+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:iterativepredict&rev=1405897398&do=diff Example: Iterative prediction of motion text/html 2026-04-07T18:51:47+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:maximally_inelastic_collision_of_two_identical_carts&rev=1775587907&do=diff Example: Maximally Inelastic Collision of Two Identical Carts Consider the opposite extreme - a maximally inelastic collision of the two identical carts, one initially at rest. That means the carts stick together (perhaps they have sticky material on their ends), and each has the same final momentum $p_{1xf} = p_{2xf}$$\vec{p}_f = \vec{p}_i + \vec{F}_{net} \Delta t$$E_f = E_i + W + Q$$K = \frac{1}{2}mv^{2} = \frac{1}{2}m(\frac{p}{m})^{2} = \frac{1}{2}(\frac{p^{2}}{m})$$${p}_{1xf} + {p}_{2xf} = … text/html 2014-10-11T06:03:36+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:mit_water_balloon_fight&rev=1413007416&do=diff Example: MIT Water Balloon Fight During the spring semester at MIT, residents of the parallel buildings of the East Campus Dorms battle one another with large sling-shots made from surgical hose mounted to window frames. Water balloons (with a mass of about 0.5kg) are placed in a pouch attached to the hose, which is then stretched nearly the width of the room (about 3.5 meters). If the hose obeys Hooke's Law, with a spring constant of 100N/m, how fast is the balloon traveling when it leaves the… text/html 2024-01-30T14:18:52+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:momentumfast&rev=1706624332&do=diff Example: Calculating the momentum of a fast-moving object An electron is observed to be moving with a velocity of $\langle -2.05\times10^7, 6.02\times10^7, 0\rangle\dfrac{m}{s}$. Determine the momentum of this electron. Setup You need to compute the momentum of this electron using the information provided and any information that you can collect or assume.$\vec{v}_e=\langle -2.05\times10^7, 6.02\times10^7, 0\rangle\dfrac{m}{s}$$c = 3.00\times10^8 \dfrac{m}{s}$$m_e = 9.11\times10^{-31} kg$$\ve… text/html 2014-07-10T14:33:52+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:momentumslow&rev=1405002832&do=diff Example: Calculating the momentum of a slow-moving object A 1971 Ford Pinto is observed to moving with a velocity of $\langle 22.35, 0, 1.06\rangle\dfrac{m}{s}$. Determine the momentum of this sweet ride. Setup You need to compute the momentum of a 1971 Ford Pinto using the information provided and any information that you can collect or assume.$\vec{v}_{car} = \langle 22.35, 0, 1.06\rangle\dfrac{m}{s}$$m_{car} = 884.05 kg$$|\vec{v}_{car}| \ll c = 3.00\times10^8 \dfrac{m}{s}$$\vec{p} = m \vec… text/html 2017-12-05T02:48:04+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:netforce&rev=1512442084&do=diff Example: Calculating the net force The fan cart is observed to experience a net force to the right. Determine the (vector) net force acting on the fan cart. Setup You need to compute the net force on the fan cart using the information provided and any information that you can collect or assume.$\dfrac{m}{s^2}$$m_{cart} = 0.3 kg$$F_{air} = 0.45 N$$\vec{F}_{net} = \sum \vec{F}_i = \vec{F}_1 + \vec{F}_2 + \dots$$F_{Earth} = m\vec{g}$$\vec{g} = \langle 0, -9.8, 0 \rangle \dfrac{m}{s^2}$$$F_{Ear… text/html 2024-01-31T16:37:16+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:positionpredict&rev=1706719036&do=diff Example: Predicting the location of a object undergoing constant velocity motion A cart is given a slight push along a near frictionless track (as shown in the video below). After the push, the cart is observed to move with a near constant velocity $\vec{v}_{cart} =\langle 1.2, 0, 0 \rangle \dfrac{m}{s}$$\vec{v}_{cart} =\langle 1.2, 0, 0 \rangle \dfrac{m}{s}$$\vec{r}_{i}$$\vec{r}_f = \vec{r}_i + \vec{v}_{avg} \Delta t$$$\vec{r}_f = \vec{r}_i + \vec{v}_{avg} \Delta t = \vec{r}_i + \vec{v}_{car… text/html 2015-09-29T16:44:24+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:predicting_the_motion_of_system_subject_to_a_spring_interaction&rev=1443545064&do=diff Example: Predicting the motion of a system that is subject to a spring interaction/Predicting the final location of an object moving under a non-constant force A spring has a relaxed length of (0.2m) and it has a spring constant of 8 N/m. Attached to the top of the spring is a block of mass (.06)kg. A force is exerted on the block to compress the spring to a total length of (0.1m). Predict the y position for the block after 0.1 second and 0.2 seconds.$L_0=0.2m$$|\vec{L}| = 0.1m$$8 N/m$$\vec{p_i… text/html 2014-11-16T08:05:32+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:relativemotion&rev=1416125132&do=diff Example: Calculating the velocity of a plane using relative measurements A Boeing 747 leaves the Detroit airport intending to head due west. The plane is experiencing a strong crosswind that is blowing toward the south, which has a wind speed of 10.0 m/s. Determine the speed of plane relative to the ground and the direction its compass should read if the pilot intends to fly due west at top speed. $|v_{a/g}| = 10.0 \dfrac{m}{s}$$\dfrac{km}{h}$$v_{p/a} = 255 \dfrac{m}{s}$$\vec{v}_{A/C} = \vec{v}… text/html 2014-11-20T16:32:33+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:rotational_angular_momentum_of_a_bicycle_wheel&rev=1416501153&do=diff Example: Rotational Angular Momentum of a Bicycle Wheel A bicycle wheel has a mass of 0.8kg and a radius of 32cm. If the wheel rotates in the xz plane, spinning clockwise when viewed from the +y axis, and making one full revolution in 0.75 seconds, what is the rotational angular momentum of the wheel?$I=MR^{2}$$\omega = \frac{2\pi}{T}$$\vec{L}_{rot} = I \vec{\omega}$$\vec{\omega}$$I$$\vec{\omega}$$\vec{L}_{rot} = I \vec{\omega}$$I$$I = MR^{2} = (0.8kg)(0.32m)^2 = 0.082 kg \cdot m^2$$\omega$$2\p… text/html 2014-11-14T07:01:39+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:rotational_kinetic_energy_and_work&rev=1415948499&do=diff Example: Rotational Kinetic Energy and Work In the figure which is in the representations section you observe that a wheel is mounted on a stationary axel, which is nearly frictionless so that the wheel turns freely. The wheel has an inner ring with mass 5 kg and radius 10 cm and an outer ring with mass 2 kg and radius 25 cm; the spokes have negligible mass. A string with negligible mass is wrapped around the outer ring and you pull on it, increasing the rotational speed of the wheel. During th… text/html 2014-10-22T04:06:56+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:sledding&rev=1413950816&do=diff Example: Sledding A little girl is riding her sled on a hill. If she starts a distance d up the hill, which makes an angle θ with the horizontal, how far will she travel along the flat snowy ground? Facts Child on incline of θ. The total mass of the sled and child = m.$$\Delta E_{system} = W_{surroundings}$$$$\Delta K + \Delta U_{g} = W_{friction}$$$$\Delta E_{system} = W_{surroundings}$$$$\Delta K + \Delta U_{g} = W_{friction}$$$$ \Delta K = 0$$$$\Delta U_{g} = W_{friction} \longrightarrow … text/html 2018-02-03T23:24:06+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:sliding_to_a_stop&rev=1517700246&do=diff Example: Sliding to a Stop You take a 3 kg metal block and slide it along the floor, where the coefficient of friction is only 0.4. You release the block with an initial velocity of $\langle 6, 0, 0\rangle m/s$. How long will it take for the block to come to a stop? How far does the block move? $\langle 6, 0, 0\rangle m/s$$\langle 0, 0, 0\rangle m/s$$\Delta \vec{p} = \vec{F}_{net} \Delta t$$ x: \Delta p_x = -\mu_k F_N\Delta t $$ y: \Delta p_y = (F_N - mg)\Delta t = 0 $$F_N$$ (F_N - mg) \Delta t… text/html 2016-03-25T15:58:41+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:statics&rev=1458921521&do=diff Example: Statics If a sign were hung like the one above, what would be the tension forces acting on both of the ropes? Setup To solve for the force of tension in both rope 1 and 2, both forces have to broken down into their x and y components, and then solve the resulting system of equations. $g=9.81m/s^{2}$$$F_{net} = \sum F = \sum F_{x} + \sum F_{y} = 0.$$$$T_{1,x} = T_{1}\sin{\alpha}$$$$T_{2,x} = T_{2}\sin{\beta}.$$$$\sum F_{x} = T_{2}\sin{\beta} - T_{1}\sin{\alpha}=0.$$$$T_{1,y} = T_{1}\… text/html 2016-03-25T16:04:59+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:statictorque&rev=1458921899&do=diff Example: Statics with Torque Find the force a person would have to apply support their friend doing a keg stand Setup Facts * Gravity will apply a negative acceleration of $9.81 m/s^2$ Lacking * Force of the person holding up the kegstandee * Torque of the person holding up their friend doing a keg stand$$\sum F = F_{net} = 0$$$$\sum \tau = \tau_{net}= 0$$$F_{k}$$F_{p}$$F_{g}$$$F_{net} = F_{k}+F_{p}-F_{g} = 0$$$$\tau = rF\sin{\theta}.$$$\theta$$\tau_{k}$$\tau_{p}$$\tau_{g}$$$\tau_{n… text/html 2014-10-11T06:09:55+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:the_jumper&rev=1413007795&do=diff Example: The Jumper After earning an 'A' in PHY 183 you land a job with ACME Bungee Jump company. They need to know what spring stiffness k_S to make the cords so that a jumper of mass 200kg will only fall 30m. The standard length of an un-stretched cord is just 10m.$$\Delta E_{system} = W_{surroundings}$$$$\Delta U_{grav} = mg(y_{f}-y_{i})$$$$\Delta U_{spring} = \dfrac{1}{2}ks({s_{f}}^2 - {s_{i}}^2)$$$$\Delta E_{system} = W_{surroundings}$$$$\Delta K_{jumper} + \Delta U_{grav} + \Delta U_{spri… text/html 2014-11-05T20:49:14+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:the_moment_of_inertia_of_a_bicycle_wheel&rev=1415220554&do=diff Example: The Moment of Inertia of a Bicycle Wheel A bicycle wheel has almost all its mass M located in the outer rim at radius R. What is the moment of inertia of the bicycle wheel about its center of mass? (Hint: It's helpful to think of dividing the wheel into the atoms it is made of and think about how much each atom contributes to the moment of inertia.)$I = m_{1}r^{2}_{\perp1}$$m_{2}r^{2}_{\perp2}$$I = m_{1}r^{2}_{\perp1}$$m_{2}r^{2}_{\perp2}$$m_{3}r^{3}_{\perp3}$$m_{4}r^{4}_{\perp4} + \c… text/html 2014-11-05T20:40:29+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:the_moment_of_inertia_of_a_diatomic_molecule&rev=1415220029&do=diff Example: The Moment of Inertia of a Diatomic Molecule What is the moment of inertia of a diatomic nitrogen molecule $N_{2}$ around its center of mass. The mass of a nitrogen atom is $2.3$ x $10^{-26}$ kg and the average distance between nuclei is $1.5$ x $10^{-10}$ m. Use the definition of moment of inertia carefully.$10^{-26}$$10^{-10}$$N_{2}$$I = m_{1}r^{2}_{\perp1}$$I = m_{1}r^{2}_{\perp1}$$m_{2}r^{2}_{\perp2}$$r_{\perp1} = r_{\perp2} = d/2$$I = M(d/2)^{2} + M(d/2)^{2} = 2M(d/2)^{2}$$m_{1}$$… text/html 2014-10-28T13:54:40+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:thermal_equilibrium&rev=1414504480&do=diff Example: Thermal Equilibrium A 300 gram block of aluminum at temperature 500 K is placed on a 650 gram block of iron at temperature 350 K in an insulated enclosure. At these temperatures the specific heat capacity of aluminum is approximately 1.0 J/K/gram, and the specific heat capacity of iron is approximately 0.42 J/K/gram. Within a few minutes the two metal blocks reach the same common temperature $T_{f}$$T_{f}$$T_{f}$$T_{f}$${\Delta E_{Al}} + {\Delta E_{Fe}} = 0$${\Delta E_{sys}} = W$$mC{\D… text/html 2014-10-02T15:50:42+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:two_students_colliding&rev=1412265042&do=diff Example: Colliding Students Two students are running to make it to class. They turn a corner and collide; coming to a complete stop. What force did they exert on each other. Facts Average mass of a person 68kg. Students come to complete stop. Choose student 1 as our system.$ \Delta t$$\Delta \vec{p}_{sys} = \vec{F}_{ext} \Delta t$$\vec{p}_{sys,f} = \vec{p}_{sys,i} + \vec{F}_{ext} \Delta t$$\vec{v}_{avg} = \dfrac{\vec{v}_{f} + \vec{v}_{i}}{2} = \dfrac{\Delta \vec{r}}{\Delta t}$${p}_{fx} = {p… text/html 2014-07-10T18:46:03+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:unitvector&rev=1405017963&do=diff Example: Calculating a unit vector You will find it useful to be able to calculate the unit vector of a given vector. For example, the force due to air resistance is defined in terms of the velocity unit vector. Determine the unit vector for the velocity vector, $\vec{v} = \langle 2, -1, 3 \rangle \dfrac{m}{s}$. Solution $$\hat{v} = \dfrac{\vec{v}}{|\vec{v}|} = \dfrac{\langle 2, -1, 3 \rangle \dfrac{m}{s}}{\sqrt{2^2+(-1)^2+3^2} \dfrac{m}{s}} = \dfrac{\langle 2, -1, 3 \rangle \dfrac{m}{s}}{\s… text/html 2014-07-10T19:08:13+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:vectordecomposition&rev=1405019293&do=diff Example: Determining vector components Determining the components of a vector in a given or established coordinate system is crucial for being able to explain or predict the motion of systems. In addition to other uses, in mechanics, decomposing vectors into their components is quite often used to determine $5\:m$$^{\circ}$$ r_x = |\vec{r}| \cos \theta$$ r_y = |\vec{r}| \sin \theta $$x$$y$$\theta = 35^{\circ}$$x$$$ r_x = |\vec{r}| \sin(\theta) = (5\:m) \sin(35^{\circ}) = 2.87\:m$$$y$$$ r_y = -|… text/html 2016-08-13T21:07:05+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:videoexample1&rev=1471122425&do=diff Example: Vector addition with a bear chase Video Part 1: You're being chased by a bear in a forest, where a marksman is waiting to shoot the bear with a tranquilizer dart. The marksman measures your velocity as 6 m/s at 43 degrees northeast. The bear's velocity is measured as 8 m/s at 30 degrees northeast. text/html 2016-08-14T00:58:37+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:videoswk3&rev=1471136317&do=diff Example: Gravitational force and Kinematic equations on the Moon Video Part 1: The diagram to the right (in the video) represents the motion of a rock launched from the surface of the moon. The rock is 2 kg, reaches a maximum height of 5 m, and is launched at an angle of 30 degrees. text/html 2016-08-14T22:47:12+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:videoswk4&rev=1471214832&do=diff Example: Chains and bonds of copper wire Video Part 1: You have a wire with a square-shaped cross-section. It's 3 m in length and made of copper (bond length = 2.57 e -10 m). The sides of the wire are 2mm wide. a) Calculate the number of chains in the wire. text/html 2016-08-14T22:52:41+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:videoswk5&rev=1471215161&do=diff Example: Two asteroids colliding in space Video Part 1: Two rocks are floating off in space. You've observed that the first rock has a mass of 3000 kg and an initial velocity of 20 m/s. The second rock has a mass of 1000 kg and no initial velocity. The two rocks collide; rock one hits rock two, leaving rock one with a velocity of 10 m/s. text/html 2016-08-17T19:02:26+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:videoswk6&rev=1471460546&do=diff Example: Change in momentum (parallel and perpendicular) of an orbit Video Part 1: An asteroid, m = 1.2 e 15 kg, is in an elliptical orbit around Earth. You observe its orbit at points A, B, C, and D. a) Draw vectors that represent the asteroid's Net Force and its Parallel & Perpendicular change in momentum at each point. text/html 2016-08-17T18:44:29+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:videoswk7&rev=1471459469&do=diff Example: Work and Friction + Ramp Video Part 1: You want to push a 40 kg box along a floor that has a coefficient of friction of .6. a) Calculate the expected friction force that will act against your motion. ---------- Part 2: You apply a force of 350 N for a distance of 20 m. How much work was done? text/html 2016-08-17T18:50:03+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:videoswk8&rev=1471459803&do=diff Example: Evaluating Potential Energy Graphs (In an Orbit) Video Part 1: Below (in the video) are two potential energy curves representing an object that is experiencing gravitational potential energy. a) Considering the 'total Energy' lines in each graph, draw Kinetic Energy curves that correspond with the potential curves. text/html 2016-08-17T19:06:47+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:videoswk9&rev=1471460807&do=diff Example: Spring Potential, Work, and Heat Exchange Video Part 1: You have a large spring that has been compressed 2 meters. It has a spring stiffness constant of 64.3 N/m. a) What is the spring potential energy of the spring? ---------- Part 2: text/html 2016-08-19T01:29:39+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:videoswk10&rev=1471570179&do=diff Example: Rotational Kinetic Energy when adding objects to the system Video Part 1: You have a rod spinning as depicted in the diagram (in the video). Its length is 2 m and its mass is 10 kg, rotating at a rate of 20 rad/s. a) What is the rotational kinetic energy of the rod? text/html 2016-08-14T22:56:00+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:videoswk11&rev=1471215360&do=diff Example: Two asteroids collide in space Video This problem is an extension of the two momentum principle videos from week 5. All four will be posted here, but only the videos labelled Part A and Part B deal with inelastic and elastic collisions. ---------- text/html 2016-08-17T18:57:51+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:videoswk12&rev=1471460271&do=diff Example: Torque of a wrench Video Part 1: Using a .1 m length wrench, you screw a bolt with 100 N of force at an unknown angle. a) If your Torque = 7 N*m, what is the angle of your force? ---------- Part 2: b) Your “friend” decides to “ text/html 2016-08-19T01:40:14+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:videoswk13&rev=1471570814&do=diff Example: Angular Momentum of a spinning woman Video Part 1: A young woman is standing on a swing, spinning at a rate of 1 rot/s. Her arms are initially straight out from her body, then she tucks them into herself, changing her rotating speed. She is 60 kg, with her arms being 7 kg altogether. text/html 2014-10-01T05:51:10+00:00 Anonymous (anonymous@undisclosed.example.com) http://msuperl.org/wikis/pcubed/doku.php?id=183_notes:examples:walking_in_a_boat&rev=1412142670&do=diff Example: Walking in a Boat A person is standing in a boat that is a length 2D + L (see diagram). If they walk a distance L, how far is the boat from the dock? Does the boat move L? Facts Boat Length: 2D + L Mass person: m Mass of boat: M Initial Momentum: Everything at rest $\vec{p_i} = 0$$\vec{p_f} = 0$$\Delta \vec{p}_{sys} = \vec{F}_{ext} \Delta t$$\vec{p}_{sys,f} = \vec{p}_{sys,i}$$\vec{r}_{cm} = \dfrac{1}{M_{tot}}\left(\sum_i m_i \vec{r}_i\right)$$\Delta \vec{p}_{sys} = \vec{F}_{ext} …