A weight with mass m hanging from a ceiling is pulled so that the string is at angle θ from the vertical and remains stationary. Draw the force diagram for the mass m, label the forces and resolve them into their x- and y-axes components. Calculate the force exerted by the hand.
To draw the force diagram for the mass, we need to consider the forces acting on it when it is in equilibrium.
1. Gravitational force (weight): This force acts vertically downward from the center of the mass and can be represented by the symbol mg, where g is the acceleration due to gravity.
2. Tension force: This force acts along the string and keeps the mass stationary. It pulls the mass upwards and can be represented by the symbol T.
3. Force exerted by the hand: This force is the force exerted by the hand to balance the weight of the mass. Let's represent it by F.
Now, let's resolve the forces into their x- and y-components:
1. Gravitational force (weight):
- Along the y-axis: This component is mg * cos(θ) and points downward.
- Along the x-axis: There is no component of weight in this direction.
2. Tension force:
- Along the y-axis: This component is T * sin(θ) and points upward.
- Along the x-axis: This component is T * cos(θ) and there is no net force in the x-direction.
3. Force exerted by the hand:
- Along the y-axis: This component is F * cos(θ) and points downward.
- Along the x-axis: This component is F * sin(θ) and points to the right.
To calculate the force exerted by the hand, we need to consider the vertical equilibrium condition. In equilibrium, the sum of the vertical forces must be zero. So,
T * sin(θ) - mg * cos(θ) - F * cos(θ) = 0
Rearranging this equation, we can solve for F:
F * cos(θ) = T * sin(θ) - mg * cos(θ)
F = (T * sin(θ) - mg * cos(θ)) / cos(θ)
Therefore, the force exerted by the hand is given by F = (T * tan(θ)) - (mg).
Note: The above calculations assume the absence of any other external forces or accelerations.