A 0.380 kg pendulum bob passes through the lowest part of its path at a speed of 2.03 m/s. What is the tension in the pendulum cable at this point if the pendulum is 83.1 cm long?
When the pendulum reaches its highest point, what angle does the cable make with the vertical?
What is the tension in the pendulum cable when the pendulum reaches its highest point?
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To find the tension in the pendulum cable when the bob is at its lowest point, we need to use the concepts of energy and forces.
First, let's find the potential energy and kinetic energy of the pendulum bob when it is at its lowest point.
1. Potential Energy (PE) at the lowest point:
The potential energy at the lowest point is zero because the reference point for potential energy is usually taken to be at the highest point.
2. Kinetic Energy (KE) at the lowest point:
The kinetic energy at the lowest point can be calculated using the formula:
KE = (1/2) * mass * velocity^2
Given:
mass (m) = 0.380 kg
velocity (v) = 2.03 m/s
Substituting the values into the formula:
KE = (1/2) * 0.380 kg * (2.03 m/s)^2
This gives us the kinetic energy at the lowest point.
Now, let's calculate the tension in the pendulum cable at the lowest point:
At the lowest point, the tension in the cable provides the centripetal force necessary to keep the pendulum bob moving in a circular path. This centripetal force is given by the formula:
Force (F) = mass * acceleration
Since the pendulum bob is moving in a horizontal circular path, the acceleration is the centripetal acceleration, which is given by:
acceleration (a) = (velocity^2) / radius
Given:
mass (m) = 0.380 kg
velocity (v) = 2.03 m/s
radius (r) = 83.1 cm = 0.831 m
Substituting the values into the formulas:
a = (v^2) / r
F = m * a
This will give us the tension in the pendulum cable at the lowest point.
To find the angle the cable makes with the vertical at the highest point, we need to consider the maximum potential energy and zero kinetic energy at that point.
1. Potential Energy (PE) at the highest point:
The potential energy at the highest point is the maximum because all the kinetic energy has been converted into potential energy. It can be calculated using the formula:
PE = mass * gravity * height
Given:
mass (m) = 0.380 kg
gravity (g) = 9.8 m/s^2
height (h) = 83.1 cm = 0.831 m
Substituting the values into the formula:
PE = m * g * h
This will give us the maximum potential energy at the highest point, and we can use it to find the angle the cable makes with the vertical.
2. Angle at the highest point:
At the highest point, the tension in the cable and the weight of the bob provide the centripetal force together. Since the bob is at equilibrium, the tension in the cable and the weight act radially inward, and the resultant force is equal to the centripetal force. We can write the equation as:
Tension + weight * cos(angle) = (mass * velocity^2) / radius
Given:
mass (m) = 0.380 kg
velocity (v) = 2.03 m/s
radius (r) = 0.831 m
weight = mass * gravity
Substituting the values into the equation:
Tension + (m * g * cos(angle)) = (m * v^2) / r
Rearranging the equation and solving for the angle:
cos(angle) = (m * v^2) / (r * g) - (Tension / (m * g))
angle = arccos(((m * v^2) / (r * g) - (Tension / (m * g))))
This will give us the angle the cable makes with the vertical at the highest point.
To find the tension in the pendulum cable at the highest point, we can use the same equation we derived above for the angle.
Rearranging the equation and solving for the tension:
Tension = (m * v^2) / (r * g) - (mass * gravity * cos(angle))
This will give us the tension in the pendulum cable at the highest point.