A simple pendulum is formed of a rope of length L = 2.2 m and a bob of mass m. When the pendulum makes an angle θ = 10° with the vertical, the speed of the bob is 2 m/s. The angular speed, θ', at the lowest position is equal to: (g = 10 m/s^2)
0.63 rad/s
1.32 rad/s
0.98 rad/s
2.93 rad/s
1.84 rad/s
at 10deg
height above hanging straight down = 2.2 (1-cos 10) = 2.2* 0.0152 =0.0334
so potential energy above hanging straight down = m g h = m*10*0.0334
= 0.334 m Joules
The kinetic energy there = (1/2) m v^2 = (1/2)m *4 = 2 m
so total energy there = 2 m + 0.334 m = 2.334 m Joules
The kinetic energy at the bottom
(1/2) m v^2 = 2.334 m
so
v^2 = 4.67
v = 2.16 m/s
omega * R = v
so
omega = 2.16 / 2.2 = 0.982 rad/s
Well, let's go swinging into this problem and see what we find!
The speed of the bob when it makes an angle of 10° with the vertical is given as 2 m/s. That's great news for the bob! It's having a swinging good time.
Now, we need to find the angular speed, θ', at the lowest position. To do that, we can use some basic knowledge about pendulums. The formula for the angular speed of a pendulum is given by θ' = √(g/L) where g is the acceleration due to gravity and L is the length of the pendulum string.
In this case, g is given as 10 m/s² and L is given as 2.2 m. Let's plug those values into the formula and do some quick math magic:
θ' = √(g/L)
θ' = √(10/2.2)
θ' = √4.545
θ' ≈ 2.13 rad/s
Uh-oh! It looks like none of the answer choices matches our calculation. Hmmm, I guess this is a tricky one. Let's try again with the closest answer choice, 1.84 rad/s.
θ' = √(g/L)
θ' = √(10/2.2)
θ' = √4.545
θ' ≈ 2.13 rad/s
Oops, it seems like I made the same mistake again! Silly me, it looks like there's something wrong with my calculations. I apologize for the inaccurate information.
Please consult with your teacher or refer to your textbook to get the correct answer. Remember, when it comes to clown bots, calculations can sometimes be a bit wobbly!
To find the angular speed (θ') at the lowest position, we can use the principle of conservation of mechanical energy.
The mechanical energy of the simple pendulum consists of two parts: potential energy and kinetic energy. At the highest point (θ = 10°), all the energy is in the form of potential energy, while at the lowest point (θ = 0°), all the energy is in the form of kinetic energy.
At the highest point (θ = 10°), the potential energy is equal to the gravitational potential energy:
PE = mgh
where m is the mass of the bob, g is the acceleration due to gravity, and h is the height of the bob above the lowest point. In this case, the height h can be calculated using the length of the rope and the angle θ:
h = L - L * cos(θ)
where L is the length of the rope.
So, the potential energy at the highest point is:
PE = mg(L - L * cos(θ))
At the lowest point (θ = 0°), all the energy is in the form of kinetic energy:
KE = (1/2) * I * θ'^2
where I is the moment of inertia of the bob and θ' is the angular speed at the lowest point.
For a simple pendulum, the moment of inertia can be approximated as:
I = m * L^2
Now, let's equate the potential energy at the highest point to the kinetic energy at the lowest point:
PE = KE
mg(L - L * cos(θ)) = (1/2) * m * L^2 * θ'^2
Canceling out the mass and multiplying through by 2 gives us:
g(L - L * cos(θ)) = L^2 * θ'^2
Rearranging the equation to solve for θ':
θ'^2 = g / L * (1 - cos(θ))
Taking the square root gives us:
θ' = √(g / L * (1 - cos(θ)))
Plugging in the values given in the question:
θ' = √(10 / 2.2 * (1 - cos(10°)))
θ' = √(10 / 2.2 * (1 - 0.985))
Calculating the expression gives us approximately:
θ' = √(10 / 2.2 * 0.015)
θ' = √(0.068181818 * 0.015)
θ' = √0.001022727
θ' ≈ 0.03197
Therefore, the angular speed (θ') at the lowest position is approximately 0.032 rad/s. None of the provided options match this value exactly.
To find the angular speed, θ', at the lowest position of the pendulum, we can use the principle of conservation of mechanical energy.
The mechanical energy of a simple pendulum consists of potential energy (PE) and kinetic energy (KE). At the highest point of the pendulum swing, all the energy is in the form of potential energy, and at the lowest point, all the energy is in the form of kinetic energy.
The potential energy at the highest point is given by PE = m * g * h, where m is the mass, g is the acceleration due to gravity (10 m/s^2), and h is the vertical height of the bob from its lowest position. Since the bob is at its highest point when it makes an angle of 10° with the vertical, the height can be calculated using the equation h = L - L * cos(θ), where L is the length of the rope (2.2 m) and θ is the angle in radians.
Thus, h = 2.2 - 2.2 * cos(θ) = 2.2 - 2.2 * cos(10°) = 2.0178 m.
The kinetic energy at the lowest point is given by KE = (1/2) * m * v^2, where v is the speed of the bob (2 m/s).
Equating the potential energy at the highest point to the kinetic energy at the lowest point, we get:
m * g * h = (1/2) * m * v^2.
We can cancel out the mass from both sides of the equation:
g * h = (1/2) * v^2.
Plugging in the given values, we have:
(10 m/s^2) * (2.0178 m) = (1/2) * (2 m/s)^2.
Simplifying the equation, we find:
20.178 = 2.
Therefore, the equation is not satisfied by any of the given values for the angular speed, θ', at the lowest position.