The Physics
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Opus in profectus

# Forces in Two Dimensions

## Problems

### practice

1. A 55 kg human cannonball is shot out the mouth of a 4.5 m cannon with a speed of 18 m/s at an angle of 60°. (Friction and air resistance are negligible in this problem.) Determine…
1. the magnitude of the acceleration of the human cannonball inside the barrel
2. the net force on the human cannonball inside the barrel
3. the weight of the human cannonball
4. the components of her weight that are parallel and perpendicular to the barrel of the cannon
5. the force on the feet of the human cannonball while she is being launched
The human cannonball leaves the mouth of the cannon and soars toward a net that is at the same height as the mouth of the cannon. Determine…
1. the magnitude and direction of the acceleration of the human cannonball in flight
2. the horizontal and vertical components of her initial velocity
3. the time she spends in the air
4. the distance from the mouth of the cannon to the center of a properly placed net
5. her maximum height above the mouth of the cannon

2. A 100 kg wooden crate rests on a wooden ramp with an adjustable angle of inclination.
1. Draw a free body diagram of the crate.
2. If the angle of the ramp is set to 10°, determine…
1. the component of the crate's weight that is perpendicular to the ramp
2. the component of the crate's weight that is parallel to the ramp
3. the normal force between the crate and the ramp
4. the static friction force between the crate and the ramp
3. At what angle will the crate just begin to slip?
4. If the angle of the ramp is set to 30°, determine…
1. the component of the crate's weight that is perpendicular to the ramp
2. the component of the crate's weight that is parallel to the ramp
3. the normal force between the crate and the ramp
4. the kinetic friction force between the crate and the ramp
5. the net force on the crate
6. the acceleration of the crate
3. A laboratory cart (m1) rests on an inclined track (θ). It is connected to a lead weight (m2) suspended vertically off the end of a pulley as shown in the diagram below. The system is released and the cart accelerates up the incline. (Assume the string and pulley contribute negligible mass to the system and that friction is kept low enough to be ignored.)

Draw a free body diagram for…
1. the laboratory cart
2. the lead weight
Determine…
1. the acceleration of the system
2. the tension in the string
3. the minimum mass needed for the lead weight to accelerate the cart up the incline
4. A pendulum can be used as an inexpensive accelerometer by a passenger in a car, airplane, roller coaster, or other vehicle. When the vehicle isn't accelerating, the pendulum will hang vertically. When the vehicle is accelerating, the pendulum will hang at an angle. Let m be the mass of the pendulum bob, be its length, a be the acceleration of the vehicle, and θ be the angle the pendulum deviates from the vertical.
1. Draw a free body diagram for the pendulum bob.
2. Derive an equation for acceleration of the vehicle in terms of the quantities given and known constants.

### numerical

1. A slingshot is made of a single piece of rubber tubing, connected to two halves of a forked stick 5.0 cm apart, with a lightweight leather pocket attached to the middle of the rubber tubing. A Bart Simpson wannabe places a 28 g stone in the pocket and pulls back on the rubber tubing until the stone is 30 cm away from the center of the gap in the forked stick. This takes 11.6 N of force.

1. Draw a free body diagram of the stone before it is released.
2. Determine the tension in the rubber tubing before the stone is released.
3. Draw a free body diagram of the stone immediately after it is released.
4. Determine the acceleration of the stone immediately after it is released.
2. A 61 kg skateboarder standing on a skateboard accelerates at a rate of 4.9 m/s2 down a 37° ramp.
1. Draw a free body diagram of the skateboarder.
2. Determine…
1. the skateboarder's weight
2. the component of the skateboarder's weight that is parallel to the ramp
3. the component of the skateboarder's weight that is perpendicular to the ramp
4. the net for acting on the skateboarder
5. the normal force of the skateboard on the skateboarder's shoes
6. the friction force between the skateboard and the skateboarder's shoes
3. What type of friction, static or kinetic, acts on the soles of the skateboarder's shoes in this problem? Explain your choice.
3. A wooden pallet loaded with 100 kilograms of canned tomatoes is placed on a wooden ramp. The ramp has an angle of inclination of 20.0°. A forklift is pushing the pallet up the ramp at constant velocity with force F that is parallel to the ramp.
1. Draw a free body diagram showing all the forces acting on the loaded pallet.
2. Determine the weight of the loaded pallet.
3. Determine the component of the pallet's weight that is parallel to the ramp.
4. Determine the component of the pallet's weight that is perpendicular to the ramp.
5. Determine the magnitude of the normal force of the pallet on the ramp.
6. Determine the magnitude of the friction force between the pallet and the ramp.
7. Determine the magnitude of the force F pushing the pallet up the ramp.
4. A block weighing 10.0 newtons is on a ramp inclined at 30.0° to the horizontal. A 3.0 newton force of friction f acts on the block as it is pulled up the ramp at constant velocity with force F, which is parallel to the ramp.
1. Draw a free body diagram showing all the forces acting on the block.
2. Determine the mass of the block.
3. Determine the component of the block's weight that is perpendicular to the ramp.
4. Determine the component of the block's weight that is parallel to the ramp.
5. Determine the magnitude of the normal force of the block on the ramp.
6. Determine the magnitude of the force F pulling the block up the ramp.
7. Determine the coefficient of friction between the block and the ramp.

### statistical

1. inclined-plane.txt
A group of physics students measured the acceleration of a laboratory cart on 120 cm long track that was inclined to several different heights. Transform the data so that a linear fit can be performed. Use the slope to determine the acceleration due to gravity.
2. wild-goose-chase.txt
American humans (Homo sapiens) aren't the only ones who love their lawns. Canada geese (Branta canadensis) love well-trimmed grass as well. The humans love it as a pedestal for their suburban homes. The geese love it as a food source and a runway. College campuses in North America often have lots of grass. When they do, they often have lots of Canada geese. They most certainly have lots of physics students and professors too. At the begining of the 21st century, the inevitable happened. A group of physics students and professors decided to chase a Canada goose across a college campus in an attempt to learn something about the mechanics of flight. Their results are compiled in the accompanying text file.
1. The Canada goose in this experiment basically flew horizontally. (These are big birds and they need a lot of space to get up to flight speed.)
1. Perform a quadratic fit on a position-time graph using the data in the text file.
2. Using your curve fit, calculate the horizontal accelertation of the bird.
2. A typical adult Canada goose has a mass of 4.7 kg. Determine the components of the force applied by the bird's wings in the…
1. horizontal and
2. vertical directions.
3. Given the values you calculated in part b, determine the…
1. magnitude and
2. direction (relative to the horizon) of the force applied by the bird's wings.
Source: The Physics of Bird Flight: An Experiment. Michael D. Mihail, Thomas F. George, Bernard J. Feldman. The Physics Teacher. Vol. 46 No. 155 (2008): 155–157.