A 2.0 mass tied to the end of a 1.5m string swings as a pendulum. At the lowest point In its swing, the speed of the mass is 4.6 m/s. What is speed of the mass when at the instant when the string makes and angle of 35 degrees with the vertical?
To find the speed of the mass at a given angle, we need to use the concepts of conservation of mechanical energy and the principles of circular motion.
First, let's consider the initial and final states of the pendulum:
1. At the lowest point, the pendulum is at its maximum potential energy and minimum kinetic energy. Therefore, the potential energy is equal to zero, and all the energy is in the form of kinetic energy.
2. At the given angle of 35 degrees with the vertical, the pendulum has a combination of potential and kinetic energy. The potential energy is given by the formula: PE = mgh, where m is the mass of the pendulum bob, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height of the bob above the lowest point.
Now, let's find the height h at the given angle of 35 degrees:
The height h can be calculated as: h = L - L*cosθ, where L is the length of the string (1.5 m) and θ is the angle (35 degrees).
h = 1.5m - 1.5m*cos(35 degrees)
h = 1.5m - 1.5m*cos(0.610865)
h ≈ 1.5m - 1.5m*0.799201
h ≈ 1.5m - 1.199802m
h ≈ 0.300198m
Now, let's calculate the potential energy:
PE = mgh
PE = 2.0kg * 9.8 m/s^2 * 0.300198m
PE ≈ 5.888 mJ (millijoules)
Since energy is conserved, we can equate the potential energy at the highest point (when the speed is maximum) to the potential energy at the given angle:
PE (highest point) = PE (given angle)
1/2 * m * v[2]^2 = m * g * h
Solving for v[2], the speed at the given angle:
v[2]^2 = 2 * g * h
v[2] = sqrt(2 * 9.8 m/s^2 * 0.300198m)
v[2] ≈ 2.152 m/s
Therefore, the speed of the mass when the string makes an angle of 35 degrees with the vertical is approximately 2.152 m/s.