a) Given a retort stand and clamp, a stout pin, a simple pendulum and a pencil, describe how you would use these apparatus to determine the centre of gravit...
a) Given a retort stand and clamp, a stout pin, a simple pendulum and a pencil, describe how you would use these apparatus to determine the centre of gravity of an irregularly shaped piece of cardboard of a moderate size.
(b) Using a suitable diagram, explain how the following can be obtained from a velocity-time graph:
(i) acceleration; (ii) total distance covered.
(c ) A body at rest is given an initial uniform acceleration of \(6.0\ \mathrm{ms}^{-2}\) for \(20\mathrm{s}\) after which the acceleration is reduced to \(4.0\ \mathrm{ms}^{-2}\) for the next \(10\mathrm{s}\). The body maintains the speed attained for \(30\mathrm{s}\).
Draw the velocity-time graph of the motion using the information given above. From the graph, calculate the:
maximum speed attained during the motion;
total distance traveled during the first \(30\ \mathrm{s}\);
average speed during the same time interval as in (ii) above.
(a) Determination of the centre of gravity of the cardboard
Make three small, well-spaced holes near the edge of the irregular cardboard.
Clamp the stout pin horizontally in the retort stand. Suspend the cardboard freely from the pin through one hole.
Hang the simple pendulum from the same pin, with its string very close to the surface of the cardboard.
Allow the cardboard and pendulum to come to rest. Trace the vertical line indicated by the pendulum string on the cardboard with the pencil.
Suspend the cardboard successively from each of the other two holes and trace the vertical line in each case.
The point of intersection of the three vertical lines is the centre of gravity of the cardboard.
Precautions: The pin must be firmly clamped; the cardboard must swing freely; and the pendulum must be at rest before each line is traced.
(b) Velocity-time graph
A velocity-time graph: acceleration is the gradient of AB, while distance is the shaded area under the graph.
(i) Acceleration
Acceleration is the gradient of the velocity-time graph. For the straight-line section joining points A and B,
\[a=\frac{\text{change in velocity}}{\text{time taken}}=\frac{v_B-v_A}{t_B-t_A}.\]
(ii) Total distance covered
The total distance covered is the area between the velocity-time graph and the time axis. Thus, for the graph shown, the distance is the area under the graph from the starting time to the final time.
(c) Velocity-time graph of the motion
For the first 20 s:
\[v_1=0+(6.0\times20)=120\ \text{m s}^{-1}.\]
For the next 10 s:
\[v_2=120+(4.0\times10)=160\ \text{m s}^{-1}.\]
The body then continues at \(160\ \text{m s}^{-1}\) for a further 30 s.
Plot and join the points (0, 0), (20, 120), (30, 160), and (60, 160) with straight lines.
The graph consists of straight-line sections through \((0,0)\), \((20,120)\), \((30,160)\), and \((60,160)\), where time is in seconds and velocity is in \(\text{m s}^{-1}\).
(i) Maximum speed
\[\boxed{160\ \text{m s}^{-1}}\]
(ii) Total distance travelled during the first 30 s
This is the area under the graph from 0 s to 30 s:
(a) Determination of the centre of gravity of the cardboard
Make three small, well-spaced holes near the edge of the irregular cardboard.
Clamp the stout pin horizontally in the retort stand. Suspend the cardboard freely from the pin through one hole.
Hang the simple pendulum from the same pin, with its string very close to the surface of the cardboard.
Allow the cardboard and pendulum to come to rest. Trace the vertical line indicated by the pendulum string on the cardboard with the pencil.
Suspend the cardboard successively from each of the other two holes and trace the vertical line in each case.
The point of intersection of the three vertical lines is the centre of gravity of the cardboard.
Precautions: The pin must be firmly clamped; the cardboard must swing freely; and the pendulum must be at rest before each line is traced.
(b) Velocity-time graph
A velocity-time graph: acceleration is the gradient of AB, while distance is the shaded area under the graph.
(i) Acceleration
Acceleration is the gradient of the velocity-time graph. For the straight-line section joining points A and B,
\[a=\frac{\text{change in velocity}}{\text{time taken}}=\frac{v_B-v_A}{t_B-t_A}.\]
(ii) Total distance covered
The total distance covered is the area between the velocity-time graph and the time axis. Thus, for the graph shown, the distance is the area under the graph from the starting time to the final time.
(c) Velocity-time graph of the motion
For the first 20 s:
\[v_1=0+(6.0\times20)=120\ \text{m s}^{-1}.\]
For the next 10 s:
\[v_2=120+(4.0\times10)=160\ \text{m s}^{-1}.\]
The body then continues at \(160\ \text{m s}^{-1}\) for a further 30 s.
Plot and join the points (0, 0), (20, 120), (30, 160), and (60, 160) with straight lines.
The graph consists of straight-line sections through \((0,0)\), \((20,120)\), \((30,160)\), and \((60,160)\), where time is in seconds and velocity is in \(\text{m s}^{-1}\).
(i) Maximum speed
\[\boxed{160\ \text{m s}^{-1}}\]
(ii) Total distance travelled during the first 30 s
This is the area under the graph from 0 s to 30 s:
