A 275g ball is resting on top of the spring that is mounted on the floor. You exert a force of 325N on the ball and it compresses the spring 44.5cm. If you release the ball from the position, how high above the equilibrium position of the spring-ball system will the ball rise?
To solve this problem, we can use the principle of conservation of mechanical energy. When you exert a force on the ball, it compresses the spring and gains potential energy. When the ball is released, this potential energy is converted into kinetic energy as the ball starts to move upwards.
The potential energy stored in the compressed spring can be calculated using the formula:
Potential Energy (PE) = 0.5 * k * x^2
Where k is the spring constant and x is the amount of compression (in meters). In this case, the amount of compression is given as 0.445 meters (44.5 cm = 0.445 m).
However, we need to find the spring constant, which is given by Hooke's Law:
Force (F) = k * x
Where F is the force applied to the spring and x is the amount of compression. Rearranging the formula, we can solve for k:
k = F / x
Plug in the given force and compression values:
k = 325N / 0.445m
k = 730.34 N/m
Now that we have the spring constant, we can calculate the potential energy:
PE = 0.5 * 730.34 N/m * (0.445m)^2
PE ≈ 73.53 J
Since energy is conserved, at the highest point, all the potential energy will be converted into gravitational potential energy. We can calculate the height above the equilibrium position using the formula:
Gravitational Potential Energy (GPE) = m * g * h
Where m is the mass of the ball, g is the acceleration due to gravity, and h is the height above the equilibrium position. In this case, the mass of the ball is given as 275g (0.275 kg) and the acceleration due to gravity is approximately 9.8 m/s^2.
Substituting the given values:
73.53 J = 0.275 kg * 9.8 m/s^2 * h
Solving for h:
h = 73.53 J / (0.275 kg * 9.8 m/s^2)
h ≈ 27.27 m
Therefore, the ball will rise approximately 27.27 meters above the equilibrium position of the spring-ball system.