The Physics
Hypertextbook
Opus in profectus

# Forces in Two Dimensions

## Problems

### practice

1. A 4.5 kg Canada goose is about to take flight. It starts from rest on the ground, but after a single step it is completely airborne. After 2.0 s of horizontal flight the bird has reached a speed of 6.0 m/s (fast enough to stay aloft, but not so fast that we need to worry about air resistance… at first).
1. Draw a free body diagram of the goose in flight.
2. Determine the following quantities for the goose in flight…
1. its acceleration
2. its weight
3. the magnitude and direction of the net force acting on it
4. the magnitude of the upward lift provided by its wings
5. the magnitude of the forward thrust provided by its wings
3. Any object moving through the air will experience air resistance. We just decided to ignore it temporarily. If we now admit that air resistance was present to some extent, how will this change the computed values of…
1. the acceleration?
2. the weight?
3. the net force?
4. the lift?
5. the thrust?
• All the measurements given in the problem are still valid for part c of this problem. The mass is still 4.5 kg and the bird still accelerates from rest to 6.0 m/s in 2.0 s.
2. A laboratory cart (m1 = 500 g) is pulled horizontally across a level track by a lead weight (m2 = 25 g) suspended vertically off the end of a pulley as shown in the diagram below. (Assume the string and pulley contribute negligible mass to the system and that friction is kept low enough to be ignored.)

1. Draw a free body diagram for…
1. the cart
2. the weight
2. Determine…
1. the acceleration of the system
2. the tension in the string
3. 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
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.

### disorganized

1. The T-38 Talon is a small (14 × 4 × 8 m), lightweight (4000 kg), twin-engine, high-altitude, supersonic jet used by various US Department of Defense groups and NASA for training purposes.
1. A T-38 requires 6670 N of thrust to fly at a constant horizontal velocity 300 m/s. Determine the following quantities for the T-38 at this moment…
1. the weight
2. the lift provided by the wings
3. the aerodynamic drag
2. The pilot has been instructed to accelerate horizontally at 0.10 g. Determine…
1. the new thrust of the engines (assuming the drag remains constant)
2. the time it takes to reach 360 m/s
2. A group of students in a physics class set up the experiment shown in the diagram below. A laboratory cart (m1 = 500 g) on a level track is connected by a horizontal string that runs over a pulley to a vertically suspended lead weight (m2 = 25 g). Friction on the cart is not negligible in this experiment. (Assume the string and pulley contribute negligible mass to the system, however.)

1. Draw a free body diagram for…
1. the lab cart
2. The students first use a cart with very sticky wheels and nothing moves. Determine…
1. the weight of the lead weight
2. the tension in the string connecting the weight to the cart
3. the friction force acting on the cart
3. The students find a small piece of debris lodged in one of the wheels and remove it. This reduces the fiction, but not to the point where it can be ignored. They perform the experiment and measure an acceleration of 0.40 m/s2. Determine…
1. the tension in the string
2. the new friction force acting on the cart
3. A laboratory cart (m1 = 500 g) rests on an inclined track (θ = 9°). It is connected to a lead weight (m2 = 100 g) suspended vertically off the end of a pulley as shown in the diagram below. (Assume the string and pulley contribute negligible mass to the system and that friction is kept low enough to be ignored.)

1. Draw a free body diagram for…
1. the laboratory cart
2. Determine…
1. the acceleration of the system (magnitude and direction)
2. the tension in the string
4. A kind of Atwood's machine is built from two cylinders of mass m1 and m2; a cylindrical pulley of mass m3 and radius r; a light, frictionless axle; and a piece of light, unstretchable string. The heavier mass m1 is held above the floor a height h and then relased from rest.
1. Draw a free body diagram showing all the forces acting on…
1. the heavier mass
2. the lighter mass
2. Determine…
1. the acceleration of the system
2. the tension in the string
3. the time it takes for the heavier mass to reach the floor
4. the speed of the system when the heavier mass hits the floor
5. A 61 kg skateboarder standing on a skateboard accelerates at a rate of 4.9 m/s2 down a 45° ramp.
1. Draw a free body diagram of the skateboarder.
2. Determine the…
1. normal force of the skateboard on the skateboarder's shoes
2. 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? Explain your choice.
6. 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 acceleration of the human cannonball inside the cannon
2. the components of her weight that are parallel and perpendicular to the barrel of the cannon
3. the force on the feet of the human cannonball while she is inside the cannon
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 horizontal and vertical components of her initial velocity
2. the time she spends in the air
3. the distance from the mouth of the cannon to the center of a properly placed net
7. 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.

### statistical

1. incline-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 the US 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 Twenty-first 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 about 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 (March 2008): 155–157.