A pendulum consists of a sphere of mass 1.5 kg attached to a light cord of length 10.9 m as in the figure below. The sphere is released from rest when the cord makes a 58 degree angle with the vertical, and the pivot at P is frictionless. The acceleration of gravity is 9.8 m/s^2 . Find the speed of the sphere when it is at the lowest point. Answer in units of m/s
What is the tension in the cord?
First, we need to find the vertical height from the initial position to the lowest point.
Let's denote the height as h, the length of the cord, as L (10.9 m), and the angle between the cord and the vertical as θ (58 degrees).
We can use the following equation to find h:
h = L - L*cos(θ)
h = 10.9 - 10.9*cos(58)
h = 10.9 - 10.9*0.5299
h = 10.9 - 5.7762
h = 5.1238 m
Next, we can find the potential energy change and consequent kinetic energy gain when the sphere reaches the lowest point.
Potential energy change (PE) is given by:
PE = m * g * h
PE = 1.5*9.8*5.1238
PE = 75.3303 J
At the lowest point, all the potential energy turns into kinetic energy. So, we can use the kinetic energy formula to find the speed (v) of the sphere:
KE = 0.5*m*v^2
75.3303 = 0.5*1.5*v^2
Now, we can solve for v:
v^2 = (75.3303*2)/1.5
v^2 = 100.4404
v = sqrt(100.4404)
v = 10.022 m/s
So, the speed of the sphere at the lowest point is 10.022 m/s.
Now, we need to find the tension in the cord.
At the lowest point, the centripetal force acting on the sphere is equal to the gravitational force. So the tension T in the cord can be found using the following equation:
T = F_c + F_g
T = m*v^2/L + m*g
Plug in the values:
T = 1.5*(10.022)^2/10.9 + 1.5*9.8
T = 13.7523 + 14.7
T = 28.4523 N
Thus, the tension in the cord is approximately 28.4523 N.
To find the tension in the cord, we can consider the forces acting on the sphere at the lowest point.
At the lowest point, the sphere is no longer at an angle and is moving horizontally. In this case, the tension in the cord will be equal to the weight of the sphere.
The weight of the sphere can be calculated using the formula:
Weight = mass * acceleration due to gravity = m * g
where m is the mass of the sphere and g is the acceleration due to gravity.
Given that the mass of the sphere is 1.5 kg and the acceleration due to gravity is 9.8 m/s^2, we can calculate the weight:
Weight = 1.5 kg * 9.8 m/s^2 = 14.7 N (newtons)
Therefore, the tension in the cord is 14.7 N.
To find the tension in the cord, you need to consider the forces acting on the sphere at the lowest point. At the lowest point, the tension in the cord will be equal to the weight of the sphere.
To calculate the weight of the sphere, you can use the formula:
Weight = mass × acceleration due to gravity.
Given:
Mass of the sphere (m) = 1.5 kg
Acceleration due to gravity (g) = 9.8 m/s^2
Weight = 1.5 kg × 9.8 m/s^2 = 14.7 N
Therefore, the tension in the cord when the sphere is at the lowest point is 14.7 N.