A vertical spring (ignore its mass), whose spring stiffness constant is 880 N/m, is attached to a table and is compressed down 0.160 m.
(a) What upward speed can it give to a 0.300 kg ball when released?
(b) How high above its original position (spring compressed) will the ball fly?
8.9m/s
To solve these problems, we can use the principles of energy conservation. We will consider both the potential energy stored in the compressed spring and the kinetic energy of the ball when it is released.
(a) To find the upward speed the spring can give to the ball when released, we can use the principle of conservation of energy. The potential energy stored in the compressed spring is given by the formula:
PE = (1/2) k x^2
where PE is the potential energy, k is the spring stiffness constant (880 N/m in this case), and x is the compression distance (0.160 m in this case).
Using this formula, we can find the potential energy stored in the spring:
PE = (1/2) * 880 N/m * (0.160 m)^2
= 11.264 J
According to the principle of conservation of energy, this potential energy will be converted into the kinetic energy of the ball when released. The kinetic energy formula is given by:
KE = (1/2) m v^2
where KE is the kinetic energy, m is the mass of the ball (0.300 kg in this case), and v is the velocity of the ball.
Since the potential energy is equal to the kinetic energy, we can equate the two formulas:
11.264 J = (1/2) * 0.300 kg * v^2
Solving for v, we get:
v^2 = (2 * 11.264 J) / 0.300 kg
= 75.093 m^2/s^2
v = sqrt(75.093 m^2/s^2)
= 8.66 m/s (approximately)
Therefore, the upward speed the spring can give to the ball when released is approximately 8.66 m/s.
(b) To find the maximum height the ball will reach above its original position when released, we can use the principle of conservation of mechanical energy. At the highest point of its trajectory, the ball will have no kinetic energy and all of its initial potential energy will be converted back into potential energy.
Knowing that the potential energy at the top of the trajectory is equal to the potential energy of the compressed spring, we can write:
PE_top = PE_initial
Where PE_top is the potential energy at the top of the trajectory and PE_initial is the initial potential energy when the spring is compressed.
The potential energy at the top of the trajectory can be calculated using the formula:
PE_top = m * g * h
where m is the mass of the ball (0.300 kg in this case), g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the maximum height above the original position.
The initial potential energy when the spring is compressed is given by:
PE_initial = (1/2) k x^2
Using the given values, we can equate these two equations:
m * g * h = (1/2) * 880 N/m * (0.160 m)^2
Solving for h, we get:
h = [(1/2) * 880 N/m * (0.160 m)^2] / (m * g)
= 0.025 m or 25 cm
Therefore, the ball will fly to a height of 0.025 m (or 25 cm) above its original position when released.