Work done by a force $F$ over a displacement $\Delta s$ is defined as the quantity $\Delta A = \vec{F}\Delta \vec{s}$.
Power is the work done per unit time: $P = \dfrac{\Delta A}{\Delta t} = F\dfrac{\Delta s}{\Delta t} =\vec{F}\cdot\vec{v}$.
Kinetic Energy
\[ \Delta A = mv\Delta v = \dfrac{m}{2}\Delta(v^2) = \dfrac{m\Delta v^2}{2}. \]
Summing all the small amounts of work $\Delta A$ during the particle's displacement, we can obtain the expression for the total work $A$ done by external forces:
\[A = \dfrac{mv^2}{2} - \dfrac{mv_0^2}{2},\]
where $v$ is the velocity of the material point at a certain moment, and $v_0$ is the initial velocity. By introducing the quantity $E_k = mv^2/2$, the resulting relation can be written as $A = \Delta E_k = E_k - E_{k0}$. The quantity $E_k$ is called kinetic energy.
Thus, an important theorem on the change in kinetic energy has been proven: the total work of all forces acting on a material point is equal to the change in the kinetic energy of the system. It is worth mentioning that we considered a material point, but this result can be generalized to an arbitrary system.
Potential Energy
Conservative (potential) forces are forces for which the work depends only on the initial and final positions and, therefore, does not depend on the path connecting the start and end points. The potential energy $E_p$ of such forces is defined as their work taken with a negative sign: $E_p = - A_\text{pot}$. Examples of potential forces: gravity, the elastic force of a spring, ...
For a body in a gravitational field $\vec{g}$ at a height $h$, the potential energy is $E_p = mgh$.
The total work of all forces (which consists of the work of potential forces $A_\text{pot}$ and non-potential forces $A_\text{diss}) is converted into the kinetic energy:
\[A_\text{pot} + A_\text{diss} = -\Delta E_p + A_\text{nonpot} = \Delta E_k \Rightarrow \Delta E_k + \Delta E_p = A_\text{nonpot}.\]
If the work of non-potential forces in the system is zero ($A_\text{diss} = 0$), then the total mechanical energy is conserved: $\Delta E_k + \Delta E_p = 0 \Rightarrow E_k + E_p = \text{const}$.
In this problem, the rolling of a metal ball down aluminum angles is investigated. Let us denote the angle of inclination of the angle to the horizontal as $\alpha$. The ball rolls along the angle without slipping: the velocities of the points of the ball in contact with the angle are zero. It follows that the friction force acting on the ball at the point of contact does no work, which means the mechanical energy in the system is conserved:
\[ E_k + E_p = \text{const}.\]For the ball, the kinetic energy consists not only of the kinetic energy of translational motion $mv^2/2$ but also of the kinetic energy of rotation. The rotational kinetic energy can be written as $\beta mv^2 / 2$, where $\beta$ is a proportionality coefficient that you need to determine in this problem.
For a U-shaped angle, the coefficient in the rotational kinetic energy is $\beta_1$, while for a standard (right-angled) one it is equal to $\beta_2$.
Assemble the setup as shown in the figure below.
Secure the U-shaped angle in the clamp of the stand so that one end of the angle is at the height of $h\approx 4.0~\text{cm}$ above the table surface. Attach masking tape to the outer surface of the bracket. Make marks on it every 10 cm.
А2 Measure the dependence of the rolling time $t$ on the distance traveled $L$. To do this, release the ball from rest and measure the time of descent to a specific mark. To improve accuracy, for each value of $L$ perform the measurement 3 times (measure the descent times $t_1,~t_2, ~t_3$) and average your results: $t = (t_1 + t_2 + t_3)/3$. Perform the measurements for 9 different values of $L$: $10~\text{cm},~20~\text{cm}, ..., ~90~\text{cm}$.
Replace the U-shaped angle with a standard (right-angled) one. Attach the masking tape to the inner surface so that the ball does not touch the tape during its descent. Make marks on the tape every 10 cm.
