(a) State the triangle law of vector addition. (b) Name the four physical quantities that are associated with the equationq of linear motion. (c) Using the ...
(b) Name the four physical quantities that are associated with the equationq of linear motion.
(c) Using the same set of axes, sketch and label two graphs to illustrate the variation of potential energy and kinetic energy with time for a body in simple harmonic motion.
(d)
A light spiral spring of force constant K lies on a horizontal frictionless surface and has one end fixed to a vertical wall. A block P of mass 2.0 kg placed against the free end of the spring is pushed a distance 5 cm towards the wall with 10J of energy as illustrated in the diagram above. The block is released and after 0.25s, it collides inelastically with a stationary block Q of mass 4.0 kg. Calculate the:
(i) value of k;
(ii) force used to compress the spring;
(iii) acceleration of the block p after release;
(iv) common speed after collision of the blocks.
(a) Triangle law of vector addition
The triangle law of vector addition states that if two vectors are represented in both magnitude and direction by the two adjacent sides of a triangle taken in order, then their resultant (sum) is represented in magnitude and direction by the third side of the triangle taken in the reverse order.
(b) Physical quantities associated with the equations of linear motion
The four physical quantities are:
Displacement (or distance), \(s\)
Velocity/speed (initial \(u\) and final \(v\))
Acceleration, \(a\)
Time, \(t\)
(c) Variation of potential energy and kinetic energy with time in S.H.M.
In simple harmonic motion the total mechanical energy \(E\) stays constant, so the potential energy (P.E.) and kinetic energy (K.E.) continuously interchange. When the body is at the equilibrium position the K.E. is maximum and the P.E. is zero; at the extreme positions the P.E. is maximum and the K.E. is zero. Both curves are \(\sin^2\)/\(\cos^2\) shapes that always add up to the constant total energy \(E\).
Kinetic and potential energy of a body in S.H.M. on the same axes: K.E. is maximum and P.E. zero at the equilibrium position, and vice versa at the extremes; at every instant the two add up to the constant total energy E.
Key features of the sketch: both curves lie between \(0\) and \(E\); K.E. starts at its maximum \(E\) (body passing through equilibrium) while P.E. starts at zero; wherever one curve peaks the other is zero, and for every instant \(\text{K.E.} + \text{P.E.} = E\).
(d) The spring-and-block problem
Given: compression \(e = 5\,\text{cm} = 0.05\,\text{m}\); energy stored \(W = 10\,\text{J}\); mass of P, \(m_P = 2.0\,\text{kg}\); mass of Q, \(m_Q = 4.0\,\text{kg}\); time after release \(t = 0.25\,\text{s}\).
(i) Value of the force constant \(k\)
The energy stored in a compressed spring is \(W = \tfrac{1}{2}k e^{2}\), so
The triangle law of vector addition states that if two vectors are represented in both magnitude and direction by the two adjacent sides of a triangle taken in order, then their resultant (sum) is represented in magnitude and direction by the third side of the triangle taken in the reverse order.
(b) Physical quantities associated with the equations of linear motion
The four physical quantities are:
Displacement (or distance), \(s\)
Velocity/speed (initial \(u\) and final \(v\))
Acceleration, \(a\)
Time, \(t\)
(c) Variation of potential energy and kinetic energy with time in S.H.M.
In simple harmonic motion the total mechanical energy \(E\) stays constant, so the potential energy (P.E.) and kinetic energy (K.E.) continuously interchange. When the body is at the equilibrium position the K.E. is maximum and the P.E. is zero; at the extreme positions the P.E. is maximum and the K.E. is zero. Both curves are \(\sin^2\)/\(\cos^2\) shapes that always add up to the constant total energy \(E\).
Kinetic and potential energy of a body in S.H.M. on the same axes: K.E. is maximum and P.E. zero at the equilibrium position, and vice versa at the extremes; at every instant the two add up to the constant total energy E.
Key features of the sketch: both curves lie between \(0\) and \(E\); K.E. starts at its maximum \(E\) (body passing through equilibrium) while P.E. starts at zero; wherever one curve peaks the other is zero, and for every instant \(\text{K.E.} + \text{P.E.} = E\).
(d) The spring-and-block problem
Given: compression \(e = 5\,\text{cm} = 0.05\,\text{m}\); energy stored \(W = 10\,\text{J}\); mass of P, \(m_P = 2.0\,\text{kg}\); mass of Q, \(m_Q = 4.0\,\text{kg}\); time after release \(t = 0.25\,\text{s}\).
(i) Value of the force constant \(k\)
The energy stored in a compressed spring is \(W = \tfrac{1}{2}k e^{2}\), so
