Question

A war-wolf or trebuchet is a device used during the middle Ages to throw rocks at castles and now sometimes used to fling large vegetables and pianos as a sport. A simple trebuchet is shown in Figure P10.47. Model it as a stiff rod of negligible mass, 3.00 m long, joining particles of mass m1 5 0.120 kg and m2 5 60.0 kg at its ends. It can turn on a frictionless, horizontal axle perpendicular to the rod and 14.0 cm from the large-mass particle. The operator releases the trebuchet from rest in a horizontal orientation. (a) Find the maximum speed that the small-mass object attains. (b) While the small-mass object is gaining speed, does it move with constant acceleration? (c) Does it move with constant tangential acceleration? (d) Does the trebuchet move with constant angular acceleration? (e) Does it have constant momentum? (f) Does the trebuchet–Earth system have constant mechanical energy?

Short answer

(a) The maximum speed that the small-mass object attains,$v_1=24.5m/s$

(b) While the small-mass object is gaining speed, it doesn’t move with constant acceleration

(c) No, it doesn’t move with constant tangential acceleration.

(d) No, the trebuchet doesn’t move with constant angular acceleration.

(e) No, it doesn’t have constant momentum.

(f) Ys, the trebuchet–Earth system have constant mechanical energy.

Step-by-step solution

Step: 1 Definition of conversion of energy:

Energy transformation, also known as energy conversion, is the process of changing energy from one form to another.

Find the maximum speed that the small-mass object attains:

(a)

Apply isolated model for energy for the two objects-Earth system. The maximum speed of the small mass occurs when the rod is in vertical position. The diagram is shown below:

Apply conversion of energy equation:

$$\begin{gathered} \Delta K+\Delta U=0 \\ \left(K_f-K_i\right)+\left(U_f-U_i\right)=0 \end{gathered}$$

Where the system is released from rest K_i=0. We define the configuration of the system before release to have zero potential energy.

$\left(\frac{1}{2}{m}_1{v}_1^2+\frac{1}{2}{m}_2{v}_2^2-0\right)+\left(m_{1}g{h}_1+m_{2}g{h}_2-0\right)=0$

Substituting the numerical value

$$\begin{gathered} \left[\frac{1}{2}(0.12)v_1^2+\frac{1}{2}\left(60\right)v_2^2\right]=\left[\right(0.12\left)\right(9.8\left)\right(2.86)+(60\left)\right(9.8\left)\right(-0.14\left)\right] \\ 0.06v_1^2+30v_2^2=78.96 ........................\left(1\right) \end{gathered}$$

The angular speed of the rod is

$$\begin{gathered} \omega =\frac{v_1}{2.86} \\ =\frac{v_2}{0.14} .......... \left(2\right) \end{gathered}$$

Solve for (V₂)

$$\begin{gathered} d{v}_2=\frac{0.14v_1}{2.86} \\ =0.0489v_1 \end{gathered}$$

Substitute for(V₂) from Equation (2) in Equation (1)

$$\begin{gathered} 0.06v_1^2+30{\left(0.0489v_1\right)}^2=78.96 \\ 0.06v_1^2+\left(30\right)(0.0489{)}^2{v}_1^2=78.96 \\ {v}_1^2\left(0.06+{0.0489}^2\right)=78.96 \end{gathered}$$

Solve for (V₁) ,

$$\begin{gathered} v_1=\sqrt{\frac{78.96}{0.06+30{(0.0489)}^2}} \\ =24.5 m/s \end{gathered}$$

So, the maximum speed is 24.5 m/s.

The small-mass object is gaining speed with constant acceleration:

(b)

No, because the acceleration is changing direction throughout the motion.

The constant tangential acceleration:

(c)

No, because the tangential velocity is changing throughout motion

The trebuchet move with constant angular acceleration:

(d)

No.

With the change in the speed of the smaller mass, tangential acceleration changes and the trebuchet moves with varying angular acceleration.

The constant momentum

(e)

No, since the speed is changing, there is a change in momentum throughout motion.

The trebuchet–Earth system have constant mechanical energy

(f)

Yes, because there is no external force acting on the system and the system is isolated.