Cos\theta is between the force and the motion.
The Y-Component of the Normal force is equal to the weight of gravity.
- Conservative Forces
- Work done by gravity is independent of path.
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Tension in Spring
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Gravity
- Non-Conservative Forces
- Friction
Potential Energy
U
- Energy of location
- Only conservative forces have potential energy!
- w is work!
delta U = U_b - U_a = -w_{a->b}
Gravitational Potential Energy
$$U = mgy + constant$$
- The constant is equal to zero and may be dropped
Spring Potential Energy
$$U = 1/2kx^2 + constant$$
- Constant is equal to zero and may be dropped
Energy has no absolute zero value - depends on reference frame/constant. Can only calculate absolute difference between U_b and U_a
Mechanical Energy
- Sum of kinetic and potential energy.
$$E = K + U$$
Change in Mechanical Energy
$$deltaE = deltaK + deltaU$$
$$deltaE = w_total + -w_conservative forces = w_non-conservative forces$$
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Change in mechanical energy is zero if there are no non-conservative forces (no friction).
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Conservative forces conserve mechanical energy.
- How Fast
- Kinetic Energy
- How High
- Potential Energy
- For both gravity and a spring: $$U = mgy + 1/2kx^2$$
Projectile Motion Problems
- K + U remains constant as projectile moves.
- As object moves up, kinetic energy gets converted into potential energy.
- Where kinetic energy equals 0 = the highest point in the trajectory.