Question

A ball weighing 60 N is swinging like a pendulum at the end of a 6.0 m rope. As the ball swings through its lowest point, its speed is measured at 5.0 m/s. What is the tension in the rope at this point?
A. 58.6 N
B. 85.4 N
C. 90.2 N
D. 97.3 N

Answer 1

Answer:

B. 85.4 N

Explanation:

The tension in the rope at the lowest point of the motion of the ball is antiparallel to the weight of the ball. By Newton's laws we construct the following equation of equilibrium:

$\Sigma F = T - W = \left(\frac{W}{g} \right)\cdot \frac{v^{2}}{L}$ (1)

Where:

$T$ - Tension in the rope, measured in newtons.

$W$ - Weight, measured in newtons.

$g$ - Gravitational acceleration, measured in meters per square second.

$v$ - Speed of the ball, measured in meters per second.

$L$ - Length of the rope, measured in meters.

If we know that $W = 60\,N$, $g = 9.807\,\frac{m}{s^{2}}$, $v = 5\,\frac{m}{s}$ and $L = 6\,m$, then the tension in the rope at this point is:

$T = W+\frac{W\cdot v^{2}}{g\cdot L}$

$T = W\cdot \left(1+\frac{v^{2}}{g\cdot L} \right)$

$T = (60\,N)\cdot \left[1+\frac{\left(5\,\frac{m}{s} \right)^{2}}{\left(9.807\,\frac{m}{s^{2}} \right)\cdot (6\,m)} \right]$

$T = 85.492\,N$

Hence, the correct answer is B.