1.7 m long pendulum is setup from a 3 m ceiling so that it is at its peak (h=.65 m), it is then released. At the bottom of the swing the string is cut (assume no engery loss during the cutting process)
1.) what is vertical velocity right before it hits the ground?
2.) how far does the pendulum bob travel away from the pendulum before it hits the ground?
3.) An 80.kg climber with a 20.0 kg pack climbs 8848 m to the t?op of mount everest. how much work did the climber exert in the climbing the mountain
When the string is cut, the bob has only horizontal velocity u. It keeps that horizontal velocity u until it hits the ground because there is no horizontal force after that.
So there are two separate problems here:
1. How far does it go at constant speed u for time t?
2. If you drop it from 3-1.7 = 1.3
meters high, how fast does it hit the ground and how long does it take, time t?
Problem 1:
(1/2)m u^2 = m g h
h = .65 m so
u = sqrt (2*9.81*.65) = 3.57 m/s for time t, the falling time
Problem 2:
Vi = 0 downward
(1/2) m v^2 = m g (1.3)
v^2 = 2 * 9.81 * 1.3
v = 5.05 m/s (That is the answer to part 1)
average speed down = (0+5.05)/2 = 2.53 m/s
so fall time t = 1.3/2.53 = .515 s
then from the horizontal problem
d = u t = 3.57 * .515 = .84 meters
3.) An 80.kg climber with a 20.0 kg pack climbs 8848 m to the t?op of mount everest. how much work did the climber exert in the climbing the mountain
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100 kg up 8848 meters
100 * 9.81 * 8848 = 8,679,888 Joules
(That is the useful work, his efficiency would make it a lot worse)
To answer the given questions, let's first understand the concepts and equations involved.
1.) The vertical velocity right before the pendulum hits the ground can be found using the concept of conservation of energy. The total mechanical energy of the pendulum is conserved throughout its motion. At its peak, the pendulum has potential energy and no kinetic energy. When it reaches the bottom, it has only kinetic energy and no potential energy.
Using the principle of conservation of energy:
Potential energy at the peak = Kinetic energy at the bottom
The potential energy is given by: PE = m * g * h, where m is the mass of the pendulum bob, g is the acceleration due to gravity, and h is the height at the peak.
The kinetic energy is given by: KE = 0.5 * m * v^2, where v is the velocity of the pendulum bob.
Since we are given the height at the peak (h = 0.65 m), we can calculate the potential energy. Furthermore, we know that at the bottom of the swing, the height is 0, so the potential energy is 0.
Setting up the equation:
m * g * h = 0.5 * m * v^2
Since mass cancels out, we can solve for v:
0.65 m * g = 0.5 * v^2
Now, substitute the value of g (acceleration due to gravity, approximately 9.8 m/s^2):
0.65 m * 9.8 m/s^2 = 0.5 * v^2
Simplifying the equation gives you the square of the velocity:
0.65 * 9.8 = 0.5 * v^2
Now solve for v by taking the square root:
v = √((0.65 * 9.8) / 0.5)
This will give you the vertical velocity right before the pendulum hits the ground.
2.) To determine how far the pendulum bob travels away from the pendulum before hitting the ground, we need to calculate the horizontal displacement.
Considering no air resistance and neglecting the size of the pendulum bob (assuming it is a point mass), the horizontal displacement can be calculated using the equation of motion for constant acceleration:
d = v * t
where d is the horizontal displacement, v is the horizontal velocity, and t is the time the bob takes to hit the ground after the string is cut.
To find the time, we first need to calculate the total time the pendulum bob takes to reach the bottom from the peak. This can be done using the equation for the time period of a simple pendulum:
T = 2π * √(l / g)
where T is the time period, l is the length of the pendulum, and g is the acceleration due to gravity.
Using the given length of the pendulum (l = 1.7 m), we can calculate T.
Substituting the value of g (approximately 9.8 m/s^2):
T = 2π * √(1.7 m / 9.8 m/s^2)
Now, divide T by 2 to get the time taken to reach the bottom from the peak.
t = T / 2
Now substitute this time value into the displacement equation:
d = v * t
This will give you the horizontal displacement of the pendulum bob.
3.) To calculate the work done by the climber in climbing Mount Everest, we can use the equation:
Work = Force * Distance * cos(θ)
In this case, the force exerted is the weight of the climber, which is given by:
Force = mass * acceleration due to gravity
Substituting the given values:
mass of climber = 80 kg
acceleration due to gravity = 9.8 m/s^2
Now, multiply the force by the distance climbed (8848 m) and take the cosine of the angle θ, which is assumed to be 0 degrees since the climber is climbing vertically.
Work = (80 kg + 20 kg) * 9.8 m/s^2 * 8848 m * cos(0)
Simplifying this equation will give you the work done by the climber in climbing Mount Everest.
Remember to always double-check the calculations and units used.