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?
To find the tension in the pendulum cable at the lowest point, we can apply the concept of conservation of mechanical energy. At the lowest point, all the potential energy is converted into kinetic energy.
Step 1: Find the potential energy at the lowest point:
Potential energy at the lowest point = m * g * h
where m is the mass of the bob, g is the acceleration due to gravity, and h is the height from the lowest point.
Here, the height h is equal to the length of the pendulum, L.
So, potential energy at the lowest point = m * g * L
Step 2: Find the kinetic energy at the lowest point:
Kinetic energy at the lowest point = (1/2) * m * v^2
where v is the velocity at the lowest point.
Step 3: Equate the potential energy and the kinetic energy at the lowest point:
m * g * L = (1/2) * m * v^2
Step 4: Solve for the tension in the pendulum cable:
The tension in the pendulum cable at the lowest point is equal to the sum of the gravitational force and the centripetal force.
Tension at the lowest point = m * g + m * v^2 / L
Now, substituting the given values:
m = 0.380 kg
g = 9.8 m/s^2
v = 2.03 m/s
L = 83.1 cm = 0.831 m
Tension at the lowest point = (0.380 kg * 9.8 m/s^2) + (0.380 kg * (2.03 m/s)^2) / 0.831 m
To find the angle the cable makes with the vertical at the highest point, we can use the conservation of mechanical energy again:
Step 1: Find the potential energy at the highest point:
Potential energy at the highest point = m * g * h
At the highest point, the height h is equal to the length of the pendulum, L.
Step 2: Find the kinetic energy at the highest point:
The bob is momentarily at rest at the highest point. Therefore, the kinetic energy is 0.
Step 3: Equate the potential energy and the kinetic energy at the highest point:
m * g * L = 0
This equation indicates that the potential energy is equal to 0 at the highest point. Therefore, the angle the cable makes with the vertical (θ) is 90 degrees.
Finally, the tension in the pendulum cable at the highest point is equal to the gravitational force only, since there is no centripetal force:
Tension at the highest point = m * g
Substituting the given values:
m = 0.380 kg
g = 9.8 m/s^2
Tension at the highest point = 0.380 kg * 9.8 m/s^2
To find the tension in the pendulum cable at the lowest point, we can use the concept of centripetal force. At the lowest point, the tension in the cable provides the centripetal force required to keep the pendulum bob moving in a circle.
We can start by calculating the gravitational force acting on the pendulum bob at the lowest point. The gravitational force is given by the equation:
F_gravity = m * g
where m is the mass of the pendulum bob and g is the acceleration due to gravity.
Given that the mass of the bob is 0.380 kg and the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the gravitational force:
F_gravity = 0.380 kg * 9.8 m/s^2 = 3.724 N
At the lowest point, the tension in the cable is equal to the sum of the gravitational force and the centripetal force, which is provided by the tension:
Tension = F_gravity + F_centripetal
Since the speed of the pendulum bob is given as 2.03 m/s at the lowest point, we can calculate the centripetal force using the equation:
F_centripetal = m * v^2 / r
where m is the mass of the pendulum bob, v is the velocity, and r is the length of the pendulum.
Plugging in the values:
F_centripetal = 0.380 kg * (2.03 m/s)^2 / 0.831 m
F_centripetal ≈ 1.877 N
Now we can find the tension at the lowest point:
Tension = F_gravity + F_centripetal
Tension = 3.724 N + 1.877 N
Tension ≈ 5.601 N
So, the tension in the pendulum cable at the lowest point is approximately 5.601 N.
Moving on to the second part of the question, when the pendulum reaches its highest point, it comes to a momentary stop. At this point, the cable makes an angle with the vertical.
The angle can be found using trigonometry. Since the length of the pendulum is given as 83.1 cm, we can use the equation:
sin(θ) = opposite/hypotenuse
The opposite side is the vertical distance from the highest point to the lowest point, which is equal to the length of the pendulum. The hypotenuse is the actual length of the pendulum.
sin(θ) = 0.831 m / 0.831 m
sin(θ) = 1
Taking the inverse sine (sin^(-1)) of both sides, we have:
θ = sin^(-1)(1)
θ = 90°
Therefore, the angle that the cable makes with the vertical at the highest point is 90 degrees.
Lastly, when the pendulum reaches its highest point, the tension in the cable becomes solely equal to the gravitational force acting on the pendulum bob. So, the tension in the cable at the highest point is equal to the weight of the bob:
Tension = m * g
Plugging in the given values:
Tension = 0.380 kg * 9.8 m/s^2 = 3.724 N
Thus, the tension in the pendulum cable when the pendulum reaches its highest point is approximately 3.724 N.
tension=mg+ma=mg+mv^2/r=you do it.
at hightest point, gain of PE= KE at bottom,
gain of PE=1/2 .380*2.03^2/.831
mgh=gain of PE
h=gainofPE/(.380*9.8)
cos Theta= (r-h) /r
tension at highest point. mg*cosTheta