Question

A spider of mass m is swinging back and forth at the end of a strand of silk of length L. During the spider's swing the strand makes a maximum angle of θ with the vertical. What is the speed of the spider at the low point of its motion, when the strand of silk is vertical? Express your answer in terms of the variables m, θ, L, and g. A spider of mass m is swinging back and forth at the end of a strand of silk of length L. During the spider's swing the strand makes a maximum angle of θ with the vertical. What is the speed of the spider at the low point of its motion, when the strand of silk is vertical?

(Express your answer in terms of the variables m, θ, L, and g.)

Answer 1

Answer 2

To find the speed of the spider at the low point of its motion, we can use the principle of conservation of mechanical energy. At the highest point of the swing, when the strand of silk makes an angle of θ with the vertical, the spider's potential energy is maximum and its kinetic energy is minimum. At the lowest point, when the strand of silk is vertical, the spider's potential energy is minimum and its kinetic energy is maximum.

Let's consider the different forms of energy involved:

1. Potential energy (PE): The potential energy of the spider at any point during the swing is given by the gravitational potential energy formula:

   PE = mgh,

   where m is the mass of the spider, g is the acceleration due to gravity, and h is the height above the reference level (lowest point).

   At the highest point, the height h is equal to L(1 - cosθ) since the vertical displacement is L and the strand makes an angle θ with the vertical. Thus, the potential energy at the highest point is mgh = mgL(1 - cosθ).

   At the lowest point, the height h is zero, so the potential energy is also zero: PE = 0.

2. Kinetic energy (KE): The kinetic energy of the spider is given by the formula:

   KE = (1/2)mv^2,

   where v is the speed of the spider.

According to the conservation of mechanical energy, the total mechanical energy at any point during the swing remains constant:

PE + KE = constant.

At the highest point, the total mechanical energy is equal to the potential energy:

mgL(1 - cosθ) + (1/2)mv^2 = mgL(1 - cosθ).

At the lowest point, the potential energy is zero, so the total mechanical energy is only kinetic energy:

0 + (1/2)mv^2 = (1/2)mv_0^2, (where v_0 is the speed at the highest point).

Simplifying the equation, we have:

(1/2)mv^2 = (1/2)mv_0^2.

Canceling out the common factors of (1/2)m, we get:

v^2 = v_0^2.

Taking the square root of both sides, we find:

v = v_0.

Therefore, the speed of the spider at the low point of its motion is equal to the speed at the highest point, which means it is independent of the height or angle. The speed is solely determined by the initial conditions and is given by:

v = √(2gL(1 - cosθ)).

So, the speed of the spider at the low point is √(2gL(1 - cosθ)).