a 139 kg crate is released from rest and hits the ground 1.62 m below after 6.80 s. What is the mass of the crate to the left of the pulley? Assume the rope and pulley are massless.
To find the mass of the crate to the left of the pulley, we need to use the principle of conservation of energy. The potential energy of the crate at the top is converted into kinetic energy as it falls. We can set up an equation using the energy conservation principle:
Potential energy at the top = Kinetic energy at the bottom.
The potential energy (PE) of an object is given by:
PE = mgh
Where:
m = mass of the object
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = height above a reference point
So, the potential energy at the top is:
PE at the top = mgh = m * 9.8 * 1.62
The kinetic energy (KE) of an object is given by:
KE = (1/2)mv^2
Where:
m = mass of the object
v = velocity of the object
In this case, the crate is released from rest, so its initial velocity (u) is 0. The final velocity (v) at the bottom can be found using the equation of motion:
v = u + at
Where:
u = initial velocity (0)
a = acceleration due to gravity (9.8 m/s^2)
t = time taken (6.80 s)
So, we can calculate the final velocity (v):
v = u + at = 0 + 9.8 * 6.80
Now, we can substitute the values into the equation for kinetic energy:
KE at the bottom = (1/2)mv^2 = (1/2) * m * (9.8 * 6.80)^2
Since the potential energy at the top is equal to the kinetic energy at the bottom, we can set up an equation:
m * 9.8 * 1.62 = (1/2) * m * (9.8 * 6.80)^2
Now, we can solve this equation to find the mass (m).