Time to travel between positions

6. One end of a light elastic string, of natural length 5 l and modulus of elasticity 20 mg , is attached to a fixed point \(A\). A particle \(P\) of mass \(2 m\) is attached to the free end of the string and \(P\) hangs freely in equilibrium at the point \(B\).

1. Find the distance \(A B\).
   (3) The particle is now pulled vertically downwards from \(B\) to the point \(C\) and released from rest. In the subsequent motion the string does not become slack.
2. Show that \(P\) moves with simple harmonic motion with centre \(B\).
3. Find the period of this motion. The greatest speed of \(P\) during this motion is \(\frac { 1 } { 5 } \sqrt { g l }\)
4. Find the amplitude of this motion. The point \(D\) is the midpoint of \(B C\) and the point \(E\) is the highest point reached by \(P\).
5. Find the time taken by \(P\) to move directly from \(D\) to \(E\).

1. A particle \(P\) moves in a straight line with simple harmonic motion between two fixed points \(A\) and \(B\). The particle performs 2 complete oscillations per second. The midpoint of \(A B\) is \(O\) and the midpoint of \(O A\) is \(C\)

The length of \(A B\) is 0.6 m .

1. Find the maximum speed of \(P\)
2. Find the time taken by \(P\) to move directly from \(O\) to \(C\)

4. \begin{figure}[h] \end{figure} In a game at a fair, a small target \(C\) moves horizontally with simple harmonic motion between the points \(A\) and \(B\), where \(A B = 4 L\). The target moves inside a box and takes 3 s to travel from \(A\) to \(B\). A player has to shoot at \(C\), but \(C\) is only visible to the player when it passes a window \(P Q\), where \(P Q = b\). The window is initially placed with \(Q\) at the point as shown in Figure 4. The target \(C\) takes 0.75 s to pass from \(Q\) to \(P\).

1. Show that \(b = ( 2 - \sqrt { 2 } ) L\).
2. Find the speed of \(C\) as it passes \(P\). \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{51510155-a8cc-4e70-8ffa-44ed35618261-4_286_1082_1327_424}
   \end{figure} For advanced players, the window \(P Q\) is moved to the centre of \(A B\) so that \(A P = Q B\), as shown in Figure 5.
3. Find the time, in seconds to 2 decimal places, taken for \(C\) to pass from \(Q\) to \(P\) in this new position.

1. A particle \(P\) moves in a straight line with simple harmonic motion about a fixed centre \(O\) with period 2 s . At time \(t\) seconds the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(t = 0\), \(v = 0\) and \(P\) is at a point \(A\) where \(O A = 0.25 \mathrm {~m}\).

Find the smallest positive value of \(t\) for which \(A P = 0.375 \mathrm {~m}\). \section*{2.} \begin{figure}[h] \end{figure} A metal ball \(B\) of mass \(m\) is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(A\). The ball \(B\) moves in a horizontal circle with centre \(O\) vertically below \(A\), as shown in Fig. 1. The string makes a constant angle \(\alpha ^ { \circ }\) with the downward vertical and \(B\) moves with constant angular speed \(\sqrt { } ( 2 g k )\), where \(k\) is a constant. The tension in the string is \(3 m g\). By modelling \(B\) as a particle. find

1. the value of \(\alpha\),
2. the length of the string.

7. A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring of natural length \(l\). The other end of the spring is attached to a fixed point \(A\). The particle is hanging freely in equilibrium at the point \(B\), where \(A B = 1.5 l\)

1. Show that the modulus of elasticity of the spring is \(2 m g\). The particle is pulled vertically downwards from \(B\) to the point \(C\), where \(A C = 1.8 \mathrm { l }\), and released from rest.
2. Show that \(P\) moves in simple harmonic motion with centre \(B\).
3. Find the greatest magnitude of the acceleration of \(P\). The midpoint of \(B C\) is \(D\). The point \(E\) lies vertically below \(A\) and \(A E = 1.2 l\)
4. Find the time taken by \(P\) to move directly from \(D\) to \(E\).

3 The point \(O\) is on the fixed line \(l\). Points \(A\) and \(B\) on \(l\) are such that \(O A = 6 \mathrm {~m}\) and \(O B = 8 \mathrm {~m}\), with \(O\) between \(A\) and \(B\). A particle \(P\) oscillates on \(l\) in simple harmonic motion with centre \(O\). When \(P\) is at \(A\) its speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and when \(P\) is at \(B\) its speed is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that the amplitude of the motion is 10 m and find the period of the motion. Find the time taken by \(P\) to travel directly from \(A\) to \(B\), through \(O\).

2 The point \(O\) is on the fixed line \(l\). Points \(A\) and \(B\) on \(l\) are such that \(O A = 0.5 \mathrm {~m}\) and \(O B = 0.75 \mathrm {~m}\), with \(A\) between \(O\) and \(B\). A particle \(P\) of mass \(m\) oscillates on \(l\) in simple harmonic motion with centre \(O\). The ratio of the kinetic energy of \(P\) when it is at \(A\) to its kinetic energy when it is at \(B\) is \(12 : 11\). Find the amplitude of the motion. Given that the greatest speed of \(P\) is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the time taken by \(P\) to travel directly from \(A\) to \(B\).

2 A particle \(P\) moves on a straight line \(A O B\) in simple harmonic motion. The centre of the motion is \(O\), and \(P\) is instantaneously at rest at \(A\) and \(B\). The point \(C\) is on the line \(A O B\), between \(A\) and \(O\), and \(C O = 10 \mathrm {~m}\). When \(P\) is at \(C\), the magnitude of its acceleration is \(0.625 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and it is moving towards \(O\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the period of the motion, in terms of \(\pi\),
2. the amplitude of the motion. The point \(M\) is the mid-point of \(O B\). Find the time that \(P\) takes to travel directly from \(C\) to \(M\).

2 A particle \(P\) moves on a straight line in simple harmonic motion. The centre of the motion is \(O\), and the amplitude of the motion is 2.5 m . The points \(L\) and \(M\) are on the line, on opposite sides of \(O\), with \(O L = 1.5 \mathrm {~m}\). The magnitudes of the accelerations of \(P\) at \(L\) and at \(M\) are in the ratio 3:4.

1. Find the distance \(O M\).
   ................................................................................................................................. .
   The time taken by \(P\) to travel directly from \(L\) to \(M\) is 2 s .
2. Find the period of the motion.
3. Find the speed of \(P\) when it passes through \(L\).

2 A particle \(P\) moves on a straight line in simple harmonic motion. The centre of the motion is \(O\). The points \(A\) and \(B\) are on the line, on opposite sides of \(O\), with \(O A = 1.6 \mathrm {~m}\) and \(O B = 1.2 \mathrm {~m}\). The ratio of the speed of \(P\) at \(A\) to its speed at \(B\) is \(3 : 4\).

1. Find the amplitude of the motion.
   The maximum speed of \(P\) during its motion is \(\frac { 1 } { 3 } \pi \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the period of the motion.
3. Find the time taken for \(P\) to travel directly from \(A\) to \(B\).

3 A particle \(P\) moves on the positive \(x\)-axis in simple harmonic motion. The centre of the motion is a distance \(d \mathrm {~m}\) from the origin \(O\), where \(0 < d < 6.5\). The points \(A\) and \(B\) are on the positive \(x\)-axis, with \(O A = 6.5 \mathrm {~m}\) and \(O B = 7.5 \mathrm {~m}\). The magnitude of the acceleration of \(P\) when it is at \(B\) is twice the magnitude of the acceleration of \(P\) when it is at \(A\).

1. Find \(d\).
   The period of the motion is \(\pi \mathrm { s }\) and the maximum acceleration of \(P\) during the motion is \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the speed of \(P\) when it is 7 m from \(O\).
3. Find the time taken by \(P\) to travel directly from \(A\) to \(B\).

2 A particle \(P\) moves on a straight line in simple harmonic motion. The centre of the motion is \(O\). The points \(A\) and \(B\) are on the line on opposite sides of \(O\) such that \(O A = 3.5 \mathrm {~m}\) and \(O B = 1 \mathrm {~m}\). The speed of \(P\) when it is at \(B\) is twice its speed when it is at \(A\). The maximum acceleration of \(P\) is \(1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the speed of \(P\) when it is at \(O\).
   \includegraphics[max width=\textwidth, alt={}, center]{2aaf3493-6509-4668-91a2-9f4708bbbb58-04_64_1566_492_328}
2. Find the time taken by \(P\) to travel directly from \(A\) to \(B\).

2 A particle \(P\) moves on a straight line in simple harmonic motion. The centre of the motion is \(O\). The points \(A\) and \(B\) are on the line on opposite sides of \(O\) such that \(O A = 3.5 \mathrm {~m}\) and \(O B = 1 \mathrm {~m}\). The speed of \(P\) when it is at \(B\) is twice its speed when it is at \(A\). The maximum acceleration of \(P\) is \(1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the speed of \(P\) when it is at \(O\).
   \includegraphics[max width=\textwidth, alt={}, center]{bbb7dbf7-1322-42d4-a0af-3850a4ea95ac-04_64_1566_492_328}
2. Find the time taken by \(P\) to travel directly from \(A\) to \(B\).

2 The point \(O\) is on the fixed line \(l\). The point \(A\) on \(l\) is such that \(O A = 3 \mathrm {~m}\). A particle \(P\) oscillates on \(l\) in simple harmonic motion with centre \(O\) and period \(\pi\) seconds. When \(P\) is at \(A\) its speed is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed of \(P\) when it is at the point \(B\) on \(l\), where \(O B = 6 \mathrm {~m}\) and \(B\) is on the same side of \(O\) as \(A\). Find, correct to 2 decimal places, the time, in seconds, taken for \(P\) to travel directly from \(A\) to \(B\).

1
\includegraphics[max width=\textwidth, alt={}, center]{58728f93-bfdb-4f76-a9b9-3a1d1592bfc9-2_125_641_262_751} The point \(C\) is on the fixed line \(l\). Points \(A\) and \(B\) on \(l\) are such that \(A C = 4 \mathrm {~m}\) and \(C B = 2 \mathrm {~m}\), with \(C\) between \(A\) and \(B\). The point \(M\) is the mid-point of \(A B\) (see diagram). A particle \(P\) of mass \(m\) oscillates between \(A\) and \(B\) in simple harmonic motion. When \(P\) is at \(C\), its speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the magnitude of the maximum acceleration of \(P\),
2. the number of complete oscillations made by \(P\) in one minute,
3. the time that \(P\) takes to travel directly from \(A\) to \(C\).

1
\includegraphics[max width=\textwidth, alt={}, center]{62d0d8cb-8f8c-4298-9705-71a735a9a4e7-2_125_641_262_751} The point \(C\) is on the fixed line \(l\). Points \(A\) and \(B\) on \(l\) are such that \(A C = 4 \mathrm {~m}\) and \(C B = 2 \mathrm {~m}\), with \(C\) between \(A\) and \(B\). The point \(M\) is the mid-point of \(A B\) (see diagram). A particle \(P\) of mass \(m\) oscillates between \(A\) and \(B\) in simple harmonic motion. When \(P\) is at \(C\), its speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the magnitude of the maximum acceleration of \(P\),
2. the number of complete oscillations made by \(P\) in one minute,
3. the time that \(P\) takes to travel directly from \(A\) to \(C\).

1
\includegraphics[max width=\textwidth, alt={}, center]{184020e1-7ff2-4172-8d33-baff963afa76-2_125_641_262_751} The point \(C\) is on the fixed line \(l\). Points \(A\) and \(B\) on \(l\) are such that \(A C = 4 \mathrm {~m}\) and \(C B = 2 \mathrm {~m}\), with \(C\) between \(A\) and \(B\). The point \(M\) is the mid-point of \(A B\) (see diagram). A particle \(P\) of mass \(m\) oscillates between \(A\) and \(B\) in simple harmonic motion. When \(P\) is at \(C\), its speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the magnitude of the maximum acceleration of \(P\),
2. the number of complete oscillations made by \(P\) in one minute,
3. the time that \(P\) takes to travel directly from \(A\) to \(C\).

3. A particle is performing simple harmonic motion along a straight line between the points \(A\) and \(B\) where \(A B = 8 \mathrm {~m}\). The period of the motion is 12 seconds.

1. Find the maximum speed of the particle in terms of \(\pi\). The points \(P\) and \(Q\) are on the line \(A B\) at distances of 3 m and 6 m respectively from \(A\).
2. Find, correct to 3 significant figures, the time it takes for the particle to travel directly from \(P\) to \(Q\).
   (6 marks)

4. A particle of mass 0.5 kg is moving on a straight line with simple harmonic motion. At time \(t = 0\) the particle is instantaneously at rest at the point \(A\). It next comes instantaneously to rest 3 seconds later at the point \(B\) where \(A B = 4 \mathrm {~m}\).

1. For the motion of the particle write down
   1. the period,
   2. the amplitude.
2. Find the maximum kinetic energy of the particle in terms of \(\pi\). The point \(C\) lies on \(A B\) at a distance of 1.2 m from \(B\).
3. Find the time it takes the particle to travel directly from \(A\) to \(C\), giving your answer in seconds correct to 2 decimal places.
   (4 marks)