91Ó°ÊÓ

In simple harmonic motion, frequency is a crucial concept. It tells us how many oscillations occur in one second. For our exercise, we have a mass attached to a spring on a frictionless horizontal surface. The mass of the object is 5.00 kg and the spring constant is 1000 N/m. To find the frequency of the motion, we first calculate the angular frequency, denoted by \( \omega \). The formula is \( \omega = \sqrt{ \dfrac{k}{m} } \). When we substitute the values, we get \( \omega = \sqrt{ \dfrac{1000}{5.00} } = \sqrt{200} = 10\sqrt{2} \, \text{rad/s} \).

Next, to find the frequency \( f \), we use the relationship \( f = \dfrac{ \omega}{2\pi} \). Substituting the angular frequency value, we get \( f = \dfrac{10\sqrt{2}}{2\pi} \). This simplifies to approximately 2.25 Hz. So, the system oscillates around 2.25 times per second.

Potential energy is the stored energy of position in this mass-spring system. When the spring is stretched or compressed, it stores elastic potential energy. To calculate the initial potential energy \( U_i \), we use:

\( U_i = \dfrac{1}{2}kx_0^2 \). Here, the given spring constant (k) is 1000 N/m and the initial displacement (\( x_0 \)) is 0.50 m.

Plugging in these values, we get:
\( U_i = \dfrac{1}{2} \times 1000 \times (0.50)^2 = 125 \text{J} \).

This means the system has 125 Joules of stored potential energy when the mass is displaced 0.50 meters from its equilibrium position.

Kinetic energy in this context is the energy of motion of the mass. When the mass moves, it possesses kinetic energy. The formula to find the initial kinetic energy \( K_i \) is: \( K_i = \dfrac{1}{2}mv_0^2 \).

Given the mass \( m \) is 5.00 kg and the initial velocity (\( v_0 \)) is 10.0 m/s, we substitute into the formula:
\( K_i = \dfrac{1}{2} \times 5.00 \times (10.0)^2 = 250 \text{J} \).

Thus, the initial kinetic energy of the object is 250 Joules.

The amplitude of oscillation represents the maximum displacement of the object from the equilibrium position during its motion. This can be determined by the total energy in the system. The total mechanical energy \( E \) is the sum of initial kinetic and potential energy:
\( E = K_i + U_i \).
Substituting the initial kinetic and potential energies, we get:
\( E = 250 + 125 = 375 \text{J} \).

To find the amplitude \( A \), we use the formula:
\( E = \dfrac{1}{2}kA^2 \).
Solving for \( A \) gives us:
\( A^2 = \dfrac{2E}{k} \rightarrow A = \sqrt{\dfrac{2 \times 375}{1000}} = \sqrt{0.75} = 0.866 \text{m} \).

This means the object oscillates with a maximum displacement of 0.866 meters from the equilibrium position.