Question

(20%) Problem 5: You are constructing a mobile out of 5 identical toy helicopters, each of mass m = 26.3 g, 4 identical sticks (each length l = 23.8 cm, and negligible mass), and thin string of negligible mass. The distance from the left-hand side of each stick to the attachment point of the sting supporting it are xị through x4 respectively, as shown in the diagram. The tensions in the supporting strings are T, through T4 as shown. You want to design the mobile so that it is in static equilibrium. T4 T3 X3 T2 T1 W 13% Part (a) Find the tension in Newtons in the string Tj. . T1 = || HOME sin() cos() tan() cotan() asin() acos() atan() acotan() sinh() cosh() tanh) cotanh() Degrees Radians (7. 8 9 E ^^^|| 4 5 6 7 1 2 3 0 VO BACKSPACE Nu sk + END DEL CLEAR Submit Hint Feedback I give up! Hints: 0% deduction per hint. Hints remaining: Feedback: 0% deduction per feedback. 13% Part (b) Find the tension in Newtons in the string T2. 13% Part (c) Find the tension in Newtons in the string T3. A 13% Part (d) Find the tension in Newtons in the string T4. 13% Part (e) Find the distance x1, in centimeters . 13% Part (f) Find the distance x2, in centimeters. 4 13% Part (g) Find the distance x3, in centimeters. 13% Part (h) Find the distance x4, in centimeters.

Answer 1

To design the mobile so that it is in static equilibrium, we need to find the tensions in the strings and the distances between the attachment points. We can solve for the tensions in the strings by using equations derived from the conditions for static equilibrium. We can also find the distances between the attachment points by rearranging and solving these equations.

Step-by-step explanation:

In order to design the mobile so that it is in static equilibrium, we need to find the tensions in the strings and the distances between the attachment points. Let's assign variables to the unknowns:

Tj represents the tension in the string Tj, where j represents the string number (1, 2, 3, or 4).

x1, x2, x3, and x4 represent the distances between the attachment points, as shown in the diagram.

Now, let's solve each part of the problem:

Part (a) Find the tension in Newtons in the string Tj.

To find the tension in each string, we can use the condition for static equilibrium. Summing up the forces in the vertical direction for each attachment point gives us the following equations:

T1 + T2 + T3 + T4 = W (equation 1)

Summing up the moments about the left-hand side of the first stick gives us:

T1 * x1 + T2 * x2 + T3 * x3 + T4 * x4 = 0 (equation 2)

From equation 1, we can express T1 in terms of T2, T3, T4, and W:

T1 = W - T2 - T3 - T4

Substituting T1 in equation 2:

(W - T2 - T3 - T4) * x1 + T2 * x2 + T3 * x3 + T4 * x4 = 0

Expanding and rearranging the terms:

x1 * W - x1 * T2 - x1 * T3 - x1 * T4 + x2 * T2 + x3 * T3 + x4 * T4 = 0

Combining the terms:

(x2 - x1) * T2 + (x3 - x1) * T3 + (x4 - x1) * T4 = x1 * W (equation 3)

We have three unknowns (T2, T3, and T4) and three equations (equations 1, 2, and 3), so we can solve for the tensions in the strings.

Part (b) Find the tension in Newtons in the string T2.

From equation 3, we can express T2 in terms of T3, T4, x1, x2, x3, x4, and W:

T2 = (x3 - x1) * T3 + (x4 - x1) * T4 + x1 * W - (x2 - x1) * T2

Part (c) Find the tension in Newtons in the string T3.

Similarly, from equation 3, we can express T3 in terms of T2, T4, x1, x2, x3, x4, and W:

T3 = (x2 - x1) * T2 + (x4 - x1) * T4 + x1 * W - (x3 - x1) * T3

Part (d) Find the tension in Newtons in the string T4.

Again, from equation 3, we can express T4 in terms of T2, T3, x1, x2, x3, x4, and W:

T4 = (x2 - x1) * T2 + (x3 - x1) * T3 + x1 * W - (x4 - x1) * T4

Part (e) Find the distance x1, in centimeters.

To find the distance x1, we can rearrange equation 3 and solve for x1:

x1 = (x2 * T2 + x3 * T3 + x4 * T4 - x1 * W) / (T2 + T3 + T4 - W)

Part (f) Find the distance x2, in centimeters.

To find the distance x2, we can use the equation T1 * x1 + T2 * x2 + T3 * x3 + T4 * x4 = 0 and solve for x2:

x2 = -(T1 * x1 + T3 * x3 + T4 * x4) / T2

Part (g) Find the distance x3, in centimeters.

To find the distance x3, we can use the same equation T1 * x1 + T2 * x2 + T3 * x3 + T4 * x4 = 0 and solve for x3:

x3 = -(T1 * x1 + T2 * x2 + T4 * x4) / T3

Part (h) Find the distance x4, in centimeters.

To find the distance x4, we can use the equation T1 * x1 + T2 * x2 + T3 * x3 + T4 * x4 = 0 and solve for x4:

x4 = -(T1 * x1 + T2 * x2 + T3 * x3) / T4

Answer 2

To design the mobile in static equilibrium, we can find the tensions and distances of the supporting strings. Parts (a) to (h) provide step-by-step instructions on how to find the tension and distance for each string.

To design the mobile so that it is in static equilibrium, we need to find the tensions in the supporting strings and the distances x1, x2, x3, and x4.

Part (a): The tension in string Tj can be found using the equilibrium equation for the vertical component of forces.

Part (b): The tension in string T2 can be found using the equilibrium equation for the horizontal component of forces.

Part (c): The tension in string T3 can be found using the equilibrium equation for the vertical component of forces.

Part (d): The tension in string T4 can be found using the equilibrium equation for the vertical component of forces.

Part (e): The distance x1 can be found using the equilibrium equation for the horizontal component of forces.

Part (f): The distance x2 can be found using the equilibrium equation for the horizontal component of forces.

Part (g): The distance x3 can be found using the equilibrium equation for the horizontal component of forces.

Part (h): The distance x4 can be found using the equilibrium equation for the horizontal component of forces.