Consider a conical pendulum with an 80.0-kg bob on a 10.0-m wire making an angle of 5.00 with the vertical as shown. Determine (a) the horizontal and vertical components of the force exerted by the wire on the pendulum and (b) the radial acceleration of the bob.

You will find the answers here:

http://en.wikipedia.org/wiki/Conical_pendulum

The forces acting on the pendulum in the vertical direction must

be in balance since the acceleration of the bob in this direction is
zero. From Newton’s second law in the y direction,
Fy Σ = T cosθ −mg = 0

Solving for the tension T gives
T = mg
cosθ
=
(80.0 kg)(9.80 m s2 )
cos 5.00°
= 787 N
In vector form,
T
= T sinθ ˆi +T cosθ ˆj
= (68.6 N)ˆi +(784 N)ˆj
(b) From Newton’s second law in the x direction,
Fx = T sinθ = mac Σ
which gives
ac = T sinθ
m
= (787 N)sin5.00°
80.0 kg
= 0.857 m/s2
toward the center of the circle.
The length of the wire is unnecessary information. We could, on
the other hand, use it to find the radius of the circle, the speed of
the bob, and the period of the motion.

The forces acting on the pendulum in the vertical direction must

be in balance since the acceleration of the bob in this direction is
zero. From Newton’s second law in the y direction,
Fy Σ = T cosθ −mg = 0

To determine the horizontal and vertical components of the force exerted by the wire on the pendulum, and the radial acceleration of the bob in a conical pendulum, we can use the equations of motion and principles of trigonometry.

First, let's find the tension in the wire. The tension is the force exerted by the wire on the pendulum. The tension can be split into two components: the horizontal component (T_h) and the vertical component (T_v).

(a) To find T_h and T_v, we can use the following equations:

T_h = T * cosθ
T_v = T * sinθ

where T is the tension in the wire and θ is the angle between the wire and the vertical.

In this case, we are given the angle θ = 5.00°. We need to convert it to radians because trigonometric functions typically work with radians.

θ_rad = θ * (π/180)
θ_rad = 5.00° * (π/180) ≈ 0.0873 rad

Now, let's find the tension T.

The force acting on the bob in the vertical direction is the weight (mg), where m is the mass of the bob (80.0 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).

T_v = mg
T_v = 80.0 kg * 9.8 m/s^2 ≈ 784 N

Now, we can find T_h and T_v using the equations mentioned earlier:

T_h = T * cosθ_rad
T_h = 784 N * cos(0.0873 rad) ≈ 783.99 N

T_v = T * sinθ_rad
T_v = 784 N * sin(0.0873 rad) ≈ 68.47 N

Therefore, the horizontal component of the force exerted by the wire on the pendulum (T_h) is approximately 783.99 N, and the vertical component (T_v) is approximately 68.47 N.

(b) The radial acceleration of the bob can be determined using the equation:

a_r = v^2 / r

where a_r is the radial acceleration, v is the instantaneous velocity of the bob, and r is the radius of the circular path (length of the wire).

In this case, we are given the length of the wire r = 10.0 m.

To find the velocity v, we can use the centripetal acceleration equation:

a_c = (v^2) / r

where a_c is the centripetal acceleration. The centripetal acceleration is equal to the radial acceleration (a_r) because the bob moves in a circular path.

Substituting the values into the centripetal acceleration equation, we can solve for v:

a_c = a_r = (v^2) / r
a_r = (v^2) / 10.0 m

Now, we need to calculate the centripetal acceleration a_c. The centripetal acceleration is the net acceleration of an object moving in a circular path and can be found using the formula:

a_c = (v^2) / r

Rearranging the formula, we have:

v^2 = a_c * r

Substituting the values, we get:

v^2 = 9.8 m/s^2 * 10.0 m
v^2 = 98 m^2/s^2

Taking the square root of both sides:

v ≈ √(98 m^2/s^2)
v ≈ 9.899 m/s

Now, we can substitute the value of v into the radial acceleration equation:

a_r = (v^2) / 10.0 m
a_r = (9.899 m/s)^2 / 10.0 m
a_r ≈ 9.8 m/s^2

Therefore, the radial acceleration of the bob (a_r) is approximately 9.8 m/s^2.