A box with a mass of m = 7.50 kg is set on top of a spring with a spring constant
of k = 500 N/m. The box is pushed down 30.0 cm and released.
(a) What is the maximum height the box reaches after being launched by the
spring?
(b) What is the instantaneous power delivered to the box by the spring when it
is only compressed by 10.0 cm?
To find the maximum height the box reaches after being launched by the spring, we can make use of the principle of conservation of mechanical energy.
The potential energy stored in the spring at maximum compression is equal to the potential energy the box has at its maximum height.
Let's start by calculating the potential energy stored in the spring at maximum compression.
(a) Maximum Height Calculation:
1. Start by finding the potential energy stored in the spring at maximum compression:
Potential energy stored in the spring = (1/2) * k * x^2
where k is the spring constant and x is the displacement (compression) of the spring.
Given:
k = 500 N/m (spring constant)
x = 30.0 cm = 0.30 m (displacement)
Potential energy stored in the spring = (1/2) * 500 N/m * (0.30 m)^2
Potential energy stored in the spring = 22.5 J
2. According to the conservation of mechanical energy, the potential energy stored in the spring at maximum compression is equal to the potential energy the box has at its maximum height.
At maximum height, the potential energy of the box is given by:
Potential energy = m * g * h
where m is the mass, g is the acceleration due to gravity, and h is the maximum height.
Given:
m = 7.50 kg (mass)
g = 9.8 m/s^2 (acceleration due to gravity)
Potential energy = 22.5 J
22.5 J = 7.50 kg * 9.8 m/s^2 * h
h = 22.5 J / (7.50 kg * 9.8 m/s^2)
h ≈ 0.305 m
Therefore, the maximum height the box reaches after being launched by the spring is approximately 0.305 meters.
(b) Instantaneous Power Calculation:
To find the instantaneous power delivered by the spring when it is only compressed by 10.0 cm, we need to differentiate the potential energy with respect to time.
1. Start by finding the potential energy stored in the spring at the given compression:
Given:
x = 10.0 cm = 0.10 m
Potential energy stored in the spring = (1/2) * 500 N/m * (0.10 m)^2
Potential energy stored in the spring = 2.5 J
2. Instantaneous power can be calculated by differentiating the potential energy with respect to time.
Instantaneous power = d(Potential energy)/dt
The potential energy of the spring can be written as:
Potential energy = (1/2) * k * x^2
Taking the derivative with respect to time (t):
d(Potential energy)/dt = (1/2) * k * 2x * dx/dt
Note: dx/dt represents the rate of change of displacement with respect to time, i.e., the velocity.
3. To further evaluate the expression, we need to find the value of dx/dt (velocity) at the given compression. Let's assume the box is released from rest and the compression is maximum when t = 0.
Given:
x = 0.10 m (displacement at t = 0)
v = dx/dt (velocity) at t = 0
From the conservation of mechanical energy:
Potential energy = Kinetic energy at maximum compression (t = 0)
(1/2) * 500 N/m * (0.10 m)^2 = (1/2) * m * v^2
2.5 J = (1/2) * 7.50 kg * v^2
v^2 = 2.5 J / (1/2 * 7.50 kg)
v ≈ 0.667 m/s
4. Substitute the values in the expression for instantaneous power:
Instantaneous power = (1/2) * 500 N/m * 2 * 0.10 m * 0.667 m/s
Instantaneous power ≈ 16.7 W
Therefore, the instantaneous power delivered to the box by the spring when it is compressed by 10.0 cm is approximately 16.7 Watts.