A small sphere of mass m is connected to the end of a cord of length r and rotates in a vertical circle about a fixed point O. The tension force exerted by the cord on the sphere is denoted by T .
What is the correct equation for the forces in the radial direction when the cord makes an angle θ with the vertical?
When the cord makes an angle θ with the vertical, the forces in the radial direction can be represented by the equation:
∑F_r = T - mg*cos(θ)
where:
- ∑F_r is the sum of the forces in the radial direction,
- T is the tension force exerted by the cord on the sphere,
- mg is the weight of the sphere (mass m multiplied by the acceleration due to gravity g),
- cos(θ) represents the component of the weight in the radial direction.
To find the equation for the forces in the radial direction when the cord makes an angle θ with the vertical, we need to consider the forces acting on the sphere at that point.
The forces in the radial direction can be broken down into two components: the gravitational force (mg) and the tension force (T). The tension force can be further decomposed into two components: the horizontal component (Tcosθ) and the vertical component (Tsinθ).
The vertical component of tension (Tsinθ) provides the centripetal force required for circular motion. It must be equal to the gravitational force (mg) at that point:
Tsinθ = mg
The horizontal component of tension (Tcosθ) provides no contribution to the radial forces because it acts perpendicular to the radial direction.
Therefore, the correct equation for the forces in the radial direction when the cord makes an angle θ with the vertical is:
Tsinθ = mg
starting with theta in the up position
F=Mv^2 -Mg CosTheta