A 0.480 kg pendulum bob passes through the lowest part of its path at a speed of 3.10 m/s. (b) When the pendulum reaches its highest point, what angle does the cable make with the vertical? The cable of the pendulum is 80 cm or .8m.
To find the angle the cable makes with the vertical when the pendulum reaches its highest point, we can use the principle of conservation of energy. The total mechanical energy of the pendulum is conserved throughout its motion.
At the lowest point of the pendulum, all of its energy is in the form of kinetic energy. Therefore, we can calculate the total mechanical energy (E) at the lowest point using the equation:
E = KE = (1/2) * m * v^2
where m is the mass of the bob and v is its velocity.
Given:
mass of the bob (m) = 0.480 kg
velocity at the lowest point (v) = 3.10 m/s
Plugging in the values:
E = (0.5) * 0.480 kg * (3.10 m/s)^2
E = 2.2416 Joules
At the highest point of the pendulum's motion, all of the energy is in the form of potential energy. We can use the equation for gravitational potential energy:
PE = m * g * h
where m is the mass, g is the acceleration due to gravity, and h is the height of the bob above its lowest point.
Given:
mass of the bob (m) = 0.480 kg
height of the bob above its lowest point (h) = 0.8 m
acceleration due to gravity (g) = 9.8 m/s^2
Plugging in the values:
PE = 0.480 kg * 9.8 m/s^2 * 0.8 m
PE = 3.7632 Joules
Since the total mechanical energy is conserved, the potential energy at the highest point (PE) should be equal to the total energy at the lowest point (E).
PE = E
3.7632 Joules = 2.2416 Joules
Therefore, there must be an additional amount of potential energy equivalent to 3.7632 - 2.2416 = 1.5216 Joules at the highest point.
The potential energy at the highest point can also be given by:
PE = m * g * h
1.5216 Joules = 0.480 kg * 9.8 m/s^2 * h
h = 1.5216 Joules / (0.480 kg * 9.8 m/s^2)
h ≈ 0.318 m
Now, we can calculate the angle that the cable makes with the vertical using the height (h) and length of the cable (L).
tan(theta) = h / L
theta = arctan(h / L)
Given:
length of the cable (L) = 0.8 m
Plugging in the values:
theta = arctan(0.318 m / 0.8 m)
theta ≈ 21.8 degrees
Therefore, the cable makes an angle of approximately 21.8 degrees with the vertical when the pendulum reaches its highest point.