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Home , Conservative force, Linear motion, Moment of inertia, Rolling

Chapter 1 Introduction Section 1 ’s Laws

dp Newton’s three laws: = F dt

For a point particle: m m · First law (Law of )[C]: It is always possible to find a p = v = r with respect to which isolated bodies move For a : linear of the CM. uniformly. This coordinate system is called an inertial reference 2 frame. d r = F dt2

The usual statement “A body remains in its state of rest or in Third law (C): If a given body A exerts a on another body B, uniform linear as long as no external force acts on it to then body B also exerts a force on body A with equal magnitude change” is not suitable. In an accelerating frame, a force-free but opposite in direction. body will appear to accelerate. Not always valid. If the Sun moved by a , then the Inertial reference frame: In which the law of inertial is valid. Any and reaction with Earth will not take on the new values until the reference frames moving with a constant wrt stationary signal reaches the Earth. This is because the Newtonian frame is a inertial reference frame. assumes instantaneous propagation of signal (action at a distance). Examples: free , a freely falling elevator. Main points: order differential equation. Second law (C): The change in the linear momentum of a body is proportional to the external force acting on the body and is in Other assumptions: the direction of the force.

2 Absolute space: Motion takes pace in the background of (1) Solve for the and velocity of a damped linear oscillator absolute space. The space is same for all observers. whose m=1, the viscous coefficient is γ. The initial condition is [x(0),v(0)]. Consequence: The length of a rod is the same for all observers. (2) Do the same for a forced undamped oscillator with initial Absolute : Evolution of physical system in “time”. Time condition is [x(0),v(0)]. Solve for both cases: ω0 ≠ ωf and ω0 = ωf moves uniformly with equal rate for all observers. (3) A spherical ball of mass m falls under in a viscous Consequence: Time interval between two events is the same fluid. Find the position and velocity of the ball as a function of for all observers. time. Assume that the mass starts at rest from a height h Mass conservation above the ground. Apply the above solution to a raindrop whose radius is 1 mm. Assume the dynamic of air to Infinite be 10−5 kg/(m s).

Examples:

Oscillator: Equation

·· · 2 m x + 2γx + ω0 x = F0 cos ωf t

(1) Linear frictionless oscillator with the initial condition is v(0) [x(0),v(0)]: x(t) = x(0)cos ωt + sin ωt ω

(2) Solve x·· = x: x(t) = x(0)cosh t + v(0)sinh t

Exercises:

3 Section 2

Work energy theorem: fields: Fields for which F ⋅ dr = 0 for any The done on a particle by a force F as it moves from point ∮ Q A to point B: closed path. For such fields, F ⋅ dr is only function of the end ∫P B W = F ⋅ dr points, but remain independent of paths. Therefore, we choose AB ∫ A P as the reference point and define U(r) as which can be rewritten as r U(r) = − F(r′) ⋅ dr′. ∫ B P B dv dr 1 [1] W = m ⋅ dt = mv2 = T − T AB ∫ dt dt [ ] B A A 2 A Therefore,

B B A where TA, TB are the of the particle at the points A [2] F ⋅ dr = F ⋅ dr − F ⋅ dr = − UB + UA and B respectively. ∫A ∫P ∫P

The above statement is work energy theorem: The difference in Eqs. [1,2] yield the kinetic energy of a particle between two points is equal to the net work done on the particle by the external during the TA + UA = TB + UB, transit. The sum of kinetic and potential is the total energy, and Conservative force fields and Potentials: the above statement is the conservation of total energy of a mechanical systems that are conservative.

4 Example: these charges. (a) Find the position(s) of equilibrium point(s). (b) Compute the potential near the equilibrium point. (c) What (1) Spring force F k x. Therefore = − is the nature of the equilibrium point? Describe the motion of x 1 1 U(x) = − (−k x)dx = k x2 − k x 2 the test particle near this point. (d) Does your analysis depend ∫ 2 2 0 x0 on the of the test charge?

Hence, the potential energy is U(x) = k x2 /2 apart from a constant, (3) What is the potential of an electric dipole? Compute the and the total energy electric field induced by it. Sketch the potential.

1 1 (4) Philip Morse modelled the potential energy of a diatomic E = m x· 2 + k x2 2 2 molecule using a potential energy function: U r D a r r 2 where r is the distance between (2) For gravitational force F = − GMm /r2 r̂ , ( ) = e{1 − exp[− ( − e)]} the atoms of the dipolar molecule re is the equilibrium bon B B 1 1 distance and De is the depth of the potential well. (a) Sketch U(B) − U(A) = − F ⋅ dr = − Frdr = − GMm − ∫A ∫A ( rB rA ) the potential. (b) Find the minima of the potential. (c) Expand the potential near its minimum and compute the of By choosing r , we deduce A = ∞ oscillation. GMm U(r) = − r

Exercises: (1) A charged particle +Q is fixed at the origin. A test charged particle +q of mass m is fired head-on towards the charged

particle +Q with velocity v∞ from ∞. Compute the shortest distance of approach between the particles.

(2) Four immovable charges of magnitude +Q are placed at the vertices of a . A test charge +q is moving in the field of

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6 Section 3 System of Particles and Conservation Laws

1. Linear Momentum fa = fa,ext + fa,int = fa,ext + ∑ fa,b b

-th particle. Centre of mass: where fa,b is the force on the a-th particle due to the b

m ∑ ara MaCM = fa = fa,ext + fa,b RCM = ∑ ∑ ∑ ∑ ∑ ma a a a b a: Particle label From Newton’s third law:

CM reference frames: Frame in which RCM = 0. fa,b + fb,a = 0

Linear momentum of a system: Therefore,

d P P = mava = ma(VCM + va′) = M VCM = PCM = f = F . ∑ ∑ dt ∑ a,ext ext a Equation of motion:

If Fext = 0, then the total linear momentum of the system P is a · d VCM P = M = m v· = f constant. This is the statement of the conservation of linear dt ∑ a a ∑ a a momentum. fa is the on the a-th particle. Example: Two (m) connected by a spring. Example Ch 13

7 If the , N, on the particle is zero, then the L of the particle is a constant.

This law is called conservation of angular momentum.

3. Angular Momentum of a System of 2. Angular Momentum of a Single Particle Particles

y The Angular momentum of a system of particles is a sum of

V angular momentum of individual particles. The angular P .m momentum of such system about a reference point is

r

L = ∑ ra × pa, . a O where denotes the position vector of the ath particle from the x ra reference point, and pa its linear momentum. Using the CM We define the angular momentum of the particle about a coordinates, r = RCM + ra′ and v = VCM + va′: reference point O, where OP = r, as

L = ∑ ra × pa a L = r × p, = m (R + r′) × (V + v′) · ∑ a CM a CM a where p = mr is the linear momentum of the particle measured in a the reference frame. The above formulas is valid in any reference = × + ′ × ′ frame, inertial or noninertial. Now. RCM PCM ∑ ra pa Therefore, d L dp dr dp = r × + × p = r × = r × F = N dt dt dt dt L = RCM × PCM + LCM = Lorbit + LCM, where where N is the torque on the particle about the reference point O. 8 LCM = ∑ ra′ × pa′ is the angular momentum of the body about the CM, The time derivative of LP is

d LP ·· = m (r − R ) × (··r − R ). dt ∑ a a P a P 4. Equation of Motion Using Angular a

Momentum ·· Using mara = fa is the net force acting on the ath particle, we obtain We assume that xy coordinate system of the figure is an inertial d LP ·· reference frame, and the reference point O belongs to it. The = (ra − RP) × (fa − maRP). dt ∑ a angular momentum of the body about O is ·· The term (ra − RP) × (−maRP) can interpreted as the torque on LO = mara × pa. ∑ ∑ a a ·· the rigid body due to the pseudo force −maRP. The force fa is a We also compute angular momentum LP about another point P sum of external and internal forces, i.e., shown in the following figure:

fa = fa,ext + ∑ fa,b m m · · , b LP = ∑ ara′ × pa′ = ∑ a(ra − RP) × (ra − RP) a a where is the external force on the th particle, while is the fa,ext a fa,b internal force on the ath particle due to the bth particle as shown y in Fig. 15.4. Therefore,

d r'a LP ·· 1 r = (r − R ) × (f − m R ) + (r′ × f + r′ × f ) a dt ∑ a P a,ext a P ∑ ∑ a a,b b b,a P a 2 a b

RP O x

9 where . According to Newton's third law, . d LP ·· ra′ = ra − RP fa,b = − fb,a = (r − R ) × f − M(R − R ) × R . or dt ∑ a P a,ext CM P P Therefore, a d LP ·· , d LP ·· 1 = Next − M(RCM − RP) × RP [3.1] = (r − R ) × (f − m R ) + (r′ − r′b ) × f dt dt ∑ a P a,ext a P ∑ ∑ a a,b a 2 a b where is the external torque acting on the rigid body about Next the point P, and the second term of the above equation is the ·· torque about the point P induced by the pseudo force −M RCM. Note that the second term vanishes for one of the following cases:

(1) RP = RCM, i.e., CM is the reference point P. This result is due to the fact that the torque induced by a pseudo force wrt the CM is zero.

·· (2) RP = 0, or the reference point P is not accelerating. Naturally, the pseudo force is zero for this case. Figure 15.4 Internal forces between particles a and b. ·· (3) (RCM − RP)||RP, a case rarely used in solving problems on rigid bodies. Now we apply the strong form of Newton's third law, according to which the action and reaction forces are not only equal and opposite, but they are also along the joining the two particles. Hence, using the CM or any non-accelerating point as a Under this condition, the total internal torque (the second term of reference, we obtain Eq. [3.1] vanishes. Therefore, d L = N . (15.3.10) dt ext

Next = 0, then L is a constant. That is, 10 If the net torque on a system of particles is zero, then the total It is important to note that computation of L and Next requires a angular momentum of the system is conserved. reference point. The reference point can be either the centre of mass or a non-accelerating point of the rigid body.

Rigid body About a Single Axis

Solving For Motion Of A Rigid Body Example 1: A cylinder rolls down an incline without slipping. Describe the motion of the cylinder.

The equation of motion for the centre of mass of the rigid body in Solution: A cylinder is down the inclined plane without an inertial frame is slipping. The forces acting on the rigid body are gravity and . The frictional force opposes the tendency to slip, hence d PCM = F (1) the frictional force is along x. dt ext − ̂ For rolling without slipping, it is convenient to use the clockwise where PCM is the linear momentum of the CM, and Fext is the rotation as positive. We denote the velocity of the CM by v, and external force acting on the rigid body. the of the cylinder along −ẑ by Ω. Note that Ω is The equation of motion for the of the rigid body is the same for all the points on the cylinder. Let a be the of the cylinder, and α its . We d L = N , (2) take the CM of the cylinder to be the reference point. We can do dt ext so even though the CM is accelerating because the pseudo force where L is the angular momentum of the rigid body about a passes through the CM, and it causes no additional torque. reference point, and Next is the external torque acting on the rigid Therefore, the equation of motion of the cylinder is body about the same reference point. ma = mg sin θ − f, (1)

Iα = f R. (2)

11 The constraint that the cylinder rolls down without slipping yields Ia k f = = mg sin θ . (6) R2 1 + k v = ωR (3) where k = I/(mR2). For the cylinder I = MR2 /2, hence k = 1/2 and By taking a time derivative of Eq. (3) we obtain a = (2g/3)sin θ.

a = αR (4) 1 X = at2, (7) CM 2

1 ϕ = αt2, (8) z 2

A block slides with an acceleration a = g sin θ.

Note that f ≤ μN, where μ is the coefficient of friction, and N is the normal force. If the frictional force cannot support rolling, then the ball will slip and roll, and we cannot apply the condition a = αR.

For rolling without slipping, the frictional force does not do any

work ( f . dr = 0) because the contact point does not slip (dr = 0). ∫

Since the frictional force does not dissipate any energy, we can Now we have three equations (1,2,4), and three unknowns a , α, also solve the above problem by applying conservation of total and f. Solving these equations yields energy:

g θ sin (5) 1 1 a = mgx sin θ = mV 2 + IΩ2, (9) 1 + k 2 CM 2 and

12 with VCM = ΩR. The LHS is the loss in potential energy, and the a = g(sin θ − μ cos θ) RHS is the gain in KE. The time derivative of the above equation α k g R θ. yields the same acceleration as Eq. (5). = ( / )cos

In the reference frame xyz, acceleration of any point on the cylinder is aP = a − αr = αR − αr, where r is the distance of the Example 2: The cylinder of Example 16.1 is pushed up the point P from the axis of the cylinder. Clearly the bottom-most incline in such a way that the cylinder rolls up the incline without point (the contact point) of the cylinder has aP = 0. As a slipping. Describe the motion of the cylinder. consequence, we can choose the bottom-most point of the Solution: When the cylinder is rolled up, the frictional force is still cylinder as the reference point since it is a non-accelerating point upward in order to oppose the tendency of the cylinder to slip of the rigid body. The equation for motion for the cylinder about down the incline. Hence the are exactly the the contact point is same as Eq. (1-4) of the above example, As a result, the cylinder I′α = (mg sin θ)R (10) has the same acceleration as Eq. (5); the cylinder slows down while ascending the incline. Note that the motion for this case is with I I MR2. Hence ′ = + the time reversed motion of Exercise 16.1. It is interesting to note g sin θ that the time-reversal symmetry is respected in this problem even α = , (11) R(1 + k) in the presence of frictional force. This is because the rolling frictional force does not dissipate energy, hence it does not break g sin θ a = , (12) the time-reversal symmetry. 1 + k which is consistent with Eq. (5).

We remark that the non-accelerating point of the cylinder at different are different. Exercises (1) A hoop of mass M and radius R is lying horizontally on a When the ball slips, f = μg cos θ. Hence smooth surface. A bug of mass m moves slowly on the hoop.

13 Compute the trajectories of the bug and the centre of mass of the (6) A charged ball of mass m, radius R and charge q is resting on hoop. a horizontal slab. We apply an electric field E on the ball. The direction of the electric field is parallel to the slab. Describe the (2) Two particles rotate around each other in a circular orbit under motion of the ball. the influence of gravitational pull of each other. The time period of orbit is T. At a given time, the particles are stopped suddenly, and then let go towards each other. After what interval of time, will the particles meet each other?

(3)The spring mass–system shown in Fig. (a) is pulled with force F. Compute the position of both the blocks as a function of time.

(4) A spring–mass system shown in Fig. (b) is resting on a horizontal surface. What force should be applied to the upper plate so that the lower one gets lifted after the pressure is removed.

(5) Two masses m1 and m2 are hanging on the two sides of a pulley that has of inertia I. Assume the string to be massless and inextensible. (a) Compute the acceleration of the masses. (b) Compute the tension of the string.

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