91Ó°ÊÓ

When studying physics, especially in spring systems, one essential concept you need to grasp is the conservation of momentum. Momentum is a measure of the motion of an object and is calculated by multiplying the object's mass and its velocity. Conservation of momentum states that in a closed system without external forces, the total momentum before and after an event remains constant.

In our exercise, two books are connected by a spring and initially stationary, meaning their total momentum is zero. When the books are released, although they start moving, the system's total momentum must still sum to zero. This conservation lets us set up the equation:

\( m_1 v_1 + m_2 v_2 = 0 \)

With this, we solve for the unknown velocity, in this case, of the physics book (because we know the speed of the calculus book). For calculations:
\[ \begin{align*} 4.0 \, kg \, v_1 + 6.0 \, kg \, \cdot \, 4.0 \, m/s &= 0 \ 4.0 \, kg \, \cdot \ v_1 &= -24.0 \, kg \, \cdot \, m/s \ v_1 &= -6.0 \, m/s \end{align*} \]

Thus, the velocity of the physics book is \( -6.0 \, m/s \).

Kinetic energy (KE) is the energy an object has due to its motion. Calculation of kinetic energy involves the mass of the object and the square of its velocity, as per the equation:
\[ K = \frac{1}{2} \, mv^2 \]

For our spring system, we know the velocities of the books obtained via momentum conservation. To find total kinetic energy, we sum the kinetic energies of both books. Here's how it's computed:
\[ \begin{align*} K_{total} &= \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 \ &= \frac{1}{2} (4.0 \, kg)(-6.0 \, m/s)^2 + \frac{1}{2} (6.0 \, kg)(4.0 \, m/s)^2 \ &= \frac{1}{2} (4.0 \, kg) (36 \, m^2 / s^2) + \frac{1}{2} (6.0 \, kg) (16 \, m^2 / s^2) \ &= 72 \, J + 48 \, J = 120 \, J \end{align*} \]

So, total kinetic energy when the spring is unstretched is 120 Joules.

Potential energy (PE) is the stored energy of an object due to its position or state. For spring systems, this involves the compression or elongation of the spring. The potential energy stored in a spring follows Hooke's Law and is calculated using the formula:
\[ E_{\text{spring}} = \frac{1}{2} k x^2 \]

However, direct calculation isn't always necessary. In our system, at the moment the spring returns to its unstretched length, all stored potential energy is converted into kinetic energy (as found in the previous section). Hence, we equate the total kinetic energy directly to the spring's potential energy:
\[ E_{\text{spring}} = K_{total} = 120 \, J \]

This tells us that 120 Joules of energy were initially stored in the spring, which was released and converted into the kinetic energy of both books. Understanding this conversion helps you grasp energy conservation principles in mechanical systems.