PHYSICS I RESOURCE SHEET

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1 PHYSICS I RESOURCE SHEET Cautions and Notes Kinematic Equations These are to be used in regions with constant acceleration only You must keep regions with different accelerations separate (for example, if a car speeds up and then slows down, treat those as two separate sets of equations) For projectile motion on Earth, we assume Forces and Newton s Laws First Law: Every object will remain at a constant velocity (zero or nonzero) unless acted upon by an outside force. Second Law: Third Law: Every force has an equal and opposite force associated with it. Steps to solving a problem using forces: Decide on the direction of the x-y axes Draw a free-body diagram Only include gravity (the only action-at-a-distance force used in Physics I) and forces due to direction contact Sum up forces in the x- and y-directions separately Apply Newton s second law in the x- and y-directions Common Forces to Consider Their Direction gravity normal force tension friction spring force down towards the Earth perpendicular to surface in the direction of the string/rope/etc. parallel to surface, opposite motion or potential motion opposite displacement of spring

2 Work and Energy Work: A force applied over a distance. Depends on component of force that is parallel to the distance. Type of Energy Formula Details Kinetic Associated with motion Gravitational Potential Associated with displacement from the Earth Elastic Potential/Spring Potential Associated with the displacement of a spring or elastic material Those types of energy together are defined as mechanical energy. Energy will remain constant (conserved) unless the system has work done on it by a non-conservative force. Non-conservative force: Path dependent, not position dependent. Friction is a good example. The only conservative forces in Physics I are gravity and the spring force; everything else is non-conservative. If there are no non-conservative forces doing work on the system, then: Momentum Momentum, p, of a system, will remain constant unless there is an outside force acting on the system. The impulse, J, of an outside force is equal to the change in momentum of the system. Type of Collision Type of Collision Type of Collision perfectly inelastic momentum objects stick together after the collision partially inelastic momentum objects do not stick together after the collision elastic momentum, kinetic energy Because both p and K are conserved, the following equation is also true:

3 Cautions and Notes: Angular Kinematic Equations Only use these equations for situations with constant angular acceleration We typically define counter-clockwise to be the positive direction, and clockwise to be the negative direction Moment of Inertia Object Moment of inertia about its Center of Mass point mass(es) hollow cylinder solid cylinder hollow sphere solid sphere long, thin beam Cautions and Notes: Moment of inertia tells us an object s resistance to angular acceleration about a particular axis For an object made up of multiple shapes (for example, a solid sphere on top of a solid cylinder), their moments of inertia can be summed For objects rotated about an axis that does not go through the center of mass (but is still parallel to an axis through the center of mass), the parallel axis theorem can be used to find the moment of inertia (where d is the distance from the center of mass):

Rotational Kinetic Energy The traditional definition of kinetic energy describes translational kinetic energy energy based on the motion of the object s center of mass!

4 Rotational Kinetic Energy The traditional definition of kinetic energy describes translational kinetic energy energy based on the motion of the object s center of mass! An object rotating in place has rotational kinetic energy! A rolling object has both translational kinetic energy and rotational kinetic energy Torque Torque is caused by a force being applied to an object at a distance from the object s center of mass. It is maximized when the force is applied perpendicular to the lever arm (the distance from the axis of rotation to the location of the force) The sum of the torques on an object is equal to that object s moment of inertia multiplied by its angular acceleration! This is often known as Newton s Second Law of Rotation Angular Momentum! Angular momentum, L, depends on an object s moment of inertia and its angular velocity. When there are no external torques applied to a system, the system s angular momentum will remain constant In other words, if

5 Practice Problems 1. On the mysterious Planet X, an alien throws a ball straight down off a cliff. The ball has an initial speed of 8 m/s (in the downward direction). The cliff is 60 m tall, and it takes 3.9 seconds for the ball to hit the ground. Air resistance is small enough that it can be ignored. What is the acceleration due to gravity on Planet X? You can ignore the effects of air resistance. 2. Frodo throws the One Ring into the fires of Mount Doom. If he throws the ring at a speed of 8 m/s at an angle of 20 degrees below the horizontal, what is the ring s speed (magnitude and direction) as it hits the lava inside Mount Doom, 100 (vertical) meters below where he threw it? You can ignore the effects of air resistance. Assume Middle Earth has the same acceleration due to gravity as regular Earth. 3. During a roller derby bout, Mouse hits Gingah Snap with a force of 200 N at an angle of 20 degrees above the horizontal. Ging (whose mass is 62 kg) has her skate wheels angled such that the force of friction between her wheels and the ground is 50 N. What is Ging s acceleration in the horizontal direction? 4. Mario jumps on a vertical spring in order to reach a coin in the video game Super Mario World. The spring has a spring constant of 9600 N/m. Mario (whose mass is 80 kg), has initially compressed the spring 26 cm from its natural length. At what speed is he moving when he leaves the spring? Assume acceleration due to gravity in Mario World is the same as it is on Earth. 5. A billiard ball, initially moving at 2.1 m/s to the right, collides with an identical billiard ball that is initially motionless. If there is no kinetic energy lost in the collision, what are the velocities of the billiard balls after the collision? 6. A record is initially spinning at 78 revolutions per minute, but then the record player is shut off and the record decreases in angular speed until it stops. It takes 4 seconds for it to slow to a stop. How many revolutions will the record go through as it slows to a stop? 7. An ice skater is spinning at a rate of 18 rad/s when she extends her arms and slows down. If her moment of inertia increases by a factor of 3 by extending her arms, what is her new angular speed?

6 Answers m/s/s m/s at an angle of 80 degrees below the horizontal m/s/s m/s 5. Initially moving ball: 0 m/s Initially stationary ball: 2.1 m/s to the right revolutions 7. 6 rad/s

PHYSICS 221, FALL 2011 EXAM #2 SOLUTIONS WEDNESDAY, NOVEMBER 2, 2011

PHYSICS 221, FALL 2011 EXAM #2 SOLUTIONS WEDNESDAY, NOVEMBER 2, 2011 PHYSICS 1, FALL 011 EXAM SOLUTIONS WEDNESDAY, NOVEMBER, 011 Note: The unit vectors in the +x, +y, and +z directions of a right-handed Cartesian coordinate system are î, ĵ, and ˆk, respectively. In this