A pendulum of length L=26.0 cm and mass m= 168 g is released from rest when the cord makes an angle of 65.2 degrees with the vertical. (a) How far (m) does the mass fall before reaching its lowest point. (b) How much work (J) is done by gravity as it falls to its lowest point? (C) How much work (J) is done by the string tension as it falls to its lowest point?
a) 26.0 cm(1 - cos65.2) = 15.1 cm
b) MgH = 0.168*9.8 *0.151 = ___ J
c) The string does zero work, because tension is always perpendicular to motion
(a) Oh dear, looks like the pendulum is having a swinging good time! To find out how far the mass falls before reaching its lowest point, we can use a bit of trigonometry. The length of the pendulum, L, gives us the radius of the circular arc that the mass traces out. Since the cord makes an angle of 65.2 degrees with the vertical, the distance fallen before reaching the lowest point would be L multiplied by the sine of that angle. So, let's do the math: L = 26.0 cm, and the angle is 65.2 degrees. We can use the formula distance fallen = L * sin(angle) to find the answer. Good luck calculating!
(b) Ah, gravity, always working hard! To find out how much work gravity does as the mass falls to its lowest point, we can use the formula work = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height fallen. But here's the trick – gravity is a conservative force, and in this case, the height is the same as the distance fallen! So, the work done by gravity is simply mgh, where m is the mass, g is the acceleration due to gravity, and h is the distance fallen. Let's crunch those numbers!
(c) Ah, the string tension, lending a helping hand! Since the pendulum is swinging freely, the string tension only acts perpendicular to the motion and doesn't do any work. So, the work done by the string tension as it falls to its lowest point is zero. Yep, you heard that right, zilch, nada, not a lot of work from the string tension in this case. Keep swinging those questions my way!
(a) To find the distance the mass falls before reaching its lowest point, we can use the conservation of energy principle:
The total mechanical energy at the highest point is given by the potential energy, which is equal to the gravitational potential energy:
Potential energy at highest point = mgh
where m is the mass, g is the acceleration due to gravity, and h is the height.
The total mechanical energy at the lowest point is given by the kinetic energy, which is equal to:
Kinetic energy at lowest point = (1/2)mv^2
where v is the velocity at the lowest point.
Since the pendulum is released from rest, the total mechanical energy is conserved, so we can equate the potential energy at the highest point to the kinetic energy at the lowest point:
mgh = (1/2)mv^2
We can cancel out the mass, m, and solve for h:
gh = (1/2)v^2
h = (1/2)(v^2/g)
To find v, we can use the equation for the velocity of a pendulum at the lowest point:
v = sqrt(2gL(1-cosθ))
where θ is the angle with the vertical (65.2 degrees in this case), L is the length of the pendulum (26.0 cm), and g is the acceleration due to gravity (9.8 m/s^2).
Converting the length to meters and the angle to radians, we can calculate the distance the mass falls before reaching its lowest point:
L = 26.0 cm = 0.26 m
θ = 65.2 degrees = 65.2 * (π/180) radians
v = sqrt(2 * 9.8 * 0.26 * (1 - cos(65.2 * π/180)))
h = (1/2)(v^2 / g)
Calculate the value of h.
(b) To find the work done by gravity as the mass falls to its lowest point, we can use the formula:
Work = mgh
where m is the mass, g is the acceleration due to gravity, and h is the height.
Using the same values for m and h as in part (a), we can calculate the work done by gravity:
Work = mgh
Calculate the value of Work.
(c) To find the work done by the string tension as the mass falls to its lowest point, we need to determine the tension at the lowest point. At the lowest point, the tension in the string must balance the gravitational force acting on the mass.
The tension in the string can be calculated using the following equation:
T = mg + ma
where T is the tension, m is the mass, g is the acceleration due to gravity, and a is the acceleration.
Since the pendulum is at its lowest point, the acceleration is equal to the centripetal acceleration, which can be calculated using the equation:
a = (v^2) / L
Using the same values for m, L, and v as in part (a), and calculating the tension T at the lowest point using the above equations.
Finally, we can calculate the work done by the string tension using the formula:
Work = T * d
where T is the tension and d is the distance.
Substituting the calculated values for T and d, we can find the work done by the string tension:
Work = T * d
Calculate the value of Work.
To find the distance the mass falls before reaching its lowest point, we can calculate the vertical displacement of the mass. The initial angle (θ) between the cord and the vertical is given as 65.2 degrees.
(a) The vertical displacement (s) of the mass can be found using the formula:
s = L * (1 - cosθ)
where L is the length of the pendulum (26.0 cm = 0.26 m) and θ is the initial angle (65.2 degrees).
Substituting the values into the formula:
s = 0.26 m * (1 - cos65.2°)
To calculate this, we need to convert the angle to radians:
θ_radians = θ_degrees * π/180
θ_radians = 65.2° * π/180 ≈ 1.1389 rad
Now we can calculate the vertical displacement:
s = 0.26 m * (1 - cos1.1389) ≈ 0.1529 m
Therefore, the mass falls approximately 0.1529 meters before reaching its lowest point.
(b) The work done by gravity as the mass falls to its lowest point can be calculated using the formula:
Work = gravitational potential energy at the highest point - gravitational potential energy at the lowest point
The gravitational potential energy (PE) at any point is given by the formula:
PE = m * g * h
where m is the mass (168 g = 0.168 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height.
At the highest point, the height (h) is equal to the length of the pendulum:
h_highest = L = 0.26 m
At the lowest point, the height (h) is equal to zero.
Substituting the values into the formula:
Work = (m * g * h_highest) - (m * g * h_lowest)
= (0.168 kg * 9.8 m/s^2 * 0.26 m) - (0.168 kg * 9.8 m/s^2 * 0)
= 0.41616 J - 0 J
= 0.41616 J
Therefore, the work done by gravity as the mass falls to its lowest point is approximately 0.41616 Joules.
(c) The work done by the string tension as the mass falls to its lowest point is zero. This is because the direction of the string tension is perpendicular to the displacement of the mass, resulting in zero work done.
Therefore, the work done by the string tension as the mass falls to its lowest point is zero Joules.