A pendulum of length L = 20.8cm and mass m = 168 g is released from rest
when the cord makes an angle of 65.2� 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) what is L(1-cos65.2) ? think why I asked that, draw the figure.
b) What is mgh
c) Isn't string tension always at 90 degrees to the movement of the mass? Work is force*distance*cosAngleBetweenThem.
To solve these problems, we will use the conservation of energy principle. The total mechanical energy of the system (mass + pendulum) remains constant throughout the motion.
(a) To find the distance the mass falls before reaching its lowest point, we need to determine the height at its highest position (initial position) and the height at its lowest point (final position).
The height at the highest position can be calculated using trigonometry:
h1 = L * cos(θ)
= 20.8 cm * cos(65.2º)
≈ 8.906 cm
The height at the lowest point is given by the length of the pendulum:
h2 = L
= 20.8 cm
The distance the mass falls is the difference between the heights:
d = h2 - h1
= 20.8 cm - 8.906 cm
≈ 11.894 cm
Therefore, the mass falls approximately 11.894 cm before reaching its lowest point.
(b) The work done by gravity as the mass falls to its lowest point can be calculated using the gravitational potential energy formula:
Gravitational Potential Energy (PEg) = m * g * h
Where:
m = mass of the pendulum (168 g = 0.168 kg)
g = acceleration due to gravity (9.8 m/s²)
h = height difference (h2 - h1) (converted to meters)
First, let's convert the heights previously calculated to meters:
h1 = 8.906 cm * 0.01 m/cm
= 0.08906 m
h2 = 20.8 cm * 0.01 m/cm
= 0.208 m
Now, we can calculate the work done by gravity:
PEg = m * g * (h2 - h1)
= 0.168 kg * 9.8 m/s² * (0.208 m - 0.08906 m)
Therefore, the work done by gravity is approximately 0.246 J.
(c) The tension in the string does work as the mass falls. Since the tension force is always perpendicular to the displacement, no work is done by the tension force. Therefore, the work done by the string tension is zero (0 J).
To solve this problem, we can utilize the principles of energy and work done by different forces. We'll start by finding the displacement and work done by gravity and tension at the lowest point.
(a) To find the distance (d) the mass falls before reaching its lowest point, we need to calculate the vertical displacement. We can do this using the formula:
d = L * (1 - cosθ)
where L is the length of the pendulum and θ is the angle with the vertical.
In this case, L = 20.8 cm = 0.208 m and θ = 65.2 degrees. Converting θ to radians:
θ = 65.2 * π/180
Now, we can calculate d:
d = 0.208 * (1 - cos(65.2 * π/180))
(b) To determine the work done by gravity as it falls to its lowest point, we'll use the formula:
Work_gravity = m * g * h
where m is the mass, g is the acceleration due to gravity, and h is the vertical displacement.
In this case, m = 168 g = 0.168 kg, g = 9.8 m/s^2, and h is the distance calculated in part (a).
Work_gravity = 0.168 * 9.8 * d
(c) To find the work done by the string tension, we need to consider that the tension force is perpendicular to the displacement. Therefore, the work done by tension is zero.
Now we can substitute the values to find the answers.