A block of mass 3 kg is hung from a spring,
causing it to stretch 21 cm at equilibrium, as
shown below.
The 3 kg block is then replaced by a 4 kg
block, and the new block is released from
the position shown below, at which point the
spring is unstretched.
21 cm
ymax
3 kg
equilbrium
4 kg
How far will the 4 kg block fall before its
direction is reversed? For the new block case,
the total distance of fall is twice the amplitude
of the oscillation. The acceleration due to
gravity is 9.8 m/s
2
.
Answer in units of cm
To find out how far the 4 kg block will fall before its direction is reversed, we need to consider the energy conservation principle.
The initial potential energy of the 4 kg block, when it is released, is equal to its gravitational potential energy at the maximum displacement. This can be calculated using the formula:
Potential energy (PE) = mass (m) * acceleration due to gravity (g) * height (h)
Since the total distance of fall is twice the amplitude of the oscillation, the height can be calculated as follows:
Height (h) = 2 * amplitude
Given that the amplitude is 21 cm, we can substitute this value into the equation:
Height (h) = 2 * 21 cm
Next, we need to calculate the potential energy of the 4 kg block using its mass and the acceleration due to gravity. Let's substitute the values into the equation:
Potential energy (PE) = 4 kg * 9.8 m/s^2 * 2 * 21 cm
Notice that we need to convert the height from centimeters to meters to match the unit of acceleration due to gravity.
1 cm = 0.01 m
Thus, the potential energy can be calculated as:
Potential energy (PE) = 4 kg * 9.8 m/s^2 * 2 * 0.21 m
To simplify the calculation, we can cancel out the units:
Potential energy (PE) = (4 * 9.8 * 2 * 0.21) kg * m^2/s^2
Now, we can simplify the expression:
Potential energy (PE) = 37.632 kg * m^2/s^2
Finally, the potential energy at the maximum displacement is the same as the initial potential energy. To find out how far the 4 kg block will fall, we need to calculate its potential energy when its direction is reversed. This energy can be found using the formula:
Potential energy (PE) = mass (m) * acceleration due to gravity (g) * height (h)
Since the height is unknown, we will denote it as ymax. The potential energy equation becomes:
Potential energy (PE) = 4 kg * 9.8 m/s^2 * ymax
From the energy conservation principle, we know that the initial potential energy is equal to the final potential energy. Therefore:
37.632 kg * m^2/s^2 = 4 kg * 9.8 m/s^2 * ymax
To find ymax, we can simplify the expression:
37.632 = 39.2 * ymax
Now, we can solve for ymax:
ymax = 37.632 / 39.2
Calculating this expression will give us the value of ymax in meters. To convert it to centimeters, we can multiply it by 100:
ymax (in cm) = ymax (in m) * 100
Now you can perform the calculations to find the value of ymax in centimeters and determine how far the 4 kg block will fall before its direction is reversed.